<?xml version="1.0" encoding="UTF-8"?><rss xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:content="http://purl.org/rss/1.0/modules/content/" xmlns:atom="http://www.w3.org/2005/Atom" version="2.0" xmlns:itunes="http://www.itunes.com/dtds/podcast-1.0.dtd" xmlns:googleplay="http://www.google.com/schemas/play-podcasts/1.0"><channel><title><![CDATA[Typal Academy: Analysis]]></title><description><![CDATA[Graphical Reasoning for Introductory Real Analysis (aka "Advanced Calculus")]]></description><link>https://newsletter.typal.academy/s/analysis</link><image><url>https://substackcdn.com/image/fetch/$s_!zJHo!,w_256,c_limit,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F725873a3-c9ed-4352-b9ea-05ef8e1d1928_400x400.png</url><title>Typal Academy: Analysis</title><link>https://newsletter.typal.academy/s/analysis</link></image><generator>Substack</generator><lastBuildDate>Mon, 06 Apr 2026 02:53:23 GMT</lastBuildDate><atom:link href="https://newsletter.typal.academy/feed" rel="self" type="application/rss+xml"/><copyright><![CDATA[Typal Academy]]></copyright><language><![CDATA[en]]></language><webMaster><![CDATA[typalacademy@substack.com]]></webMaster><itunes:owner><itunes:email><![CDATA[typalacademy@substack.com]]></itunes:email><itunes:name><![CDATA[Howard Heaton]]></itunes:name></itunes:owner><itunes:author><![CDATA[Howard Heaton]]></itunes:author><googleplay:owner><![CDATA[typalacademy@substack.com]]></googleplay:owner><googleplay:email><![CDATA[typalacademy@substack.com]]></googleplay:email><googleplay:author><![CDATA[Howard Heaton]]></googleplay:author><itunes:block><![CDATA[Yes]]></itunes:block><item><title><![CDATA[Series Divergence via Asymptotic Behavior]]></title><description><![CDATA[An example with the harmonic series]]></description><link>https://newsletter.typal.academy/p/series-divergence-via-asymptotic</link><guid isPermaLink="false">https://newsletter.typal.academy/p/series-divergence-via-asymptotic</guid><dc:creator><![CDATA[Howard Heaton]]></dc:creator><pubDate>Thu, 08 May 2025 15:12:57 GMT</pubDate><enclosure url="https://substackcdn.com/image/fetch/$s_!B0OS!,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F1d9b3e62-733e-4c31-87de-4612cf372083_1342x1228.png" length="0" type="image/jpeg"/><content:encoded><![CDATA[
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   ]]></content:encoded></item><item><title><![CDATA[Applying Definition of Supremum]]></title><description><![CDATA[If a point in a set is an upper bound, it's the supremum]]></description><link>https://newsletter.typal.academy/p/applying-definition-of-supremum</link><guid isPermaLink="false">https://newsletter.typal.academy/p/applying-definition-of-supremum</guid><dc:creator><![CDATA[Howard Heaton]]></dc:creator><pubDate>Wed, 05 Feb 2025 15:03:05 GMT</pubDate><enclosure url="https://substackcdn.com/image/fetch/$s_!kA4e!,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F70ec36d0-5fdb-44b2-a3e9-a14bb4c983f5_1764x1498.png" length="0" type="image/jpeg"/><content:encoded><![CDATA[
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   ]]></content:encoded></item><item><title><![CDATA[Distance to Set is Continuous]]></title><description><![CDATA[An &#949;-&#948; argument for uniform continuity]]></description><link>https://newsletter.typal.academy/p/distance-to-set-is-continuous</link><guid isPermaLink="false">https://newsletter.typal.academy/p/distance-to-set-is-continuous</guid><dc:creator><![CDATA[Howard Heaton]]></dc:creator><pubDate>Tue, 12 Nov 2024 17:32:34 GMT</pubDate><enclosure url="https://substackcdn.com/image/fetch/$s_!1dus!,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F39f94197-2c71-4667-af2c-9c902649fe17_1920x1920.png" length="0" type="image/jpeg"/><content:encoded><![CDATA[
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   ]]></content:encoded></item><item><title><![CDATA[Proving Two Sets are Equal]]></title><description><![CDATA[How to make a subset argument]]></description><link>https://newsletter.typal.academy/p/proving-two-sets-are-equal</link><guid isPermaLink="false">https://newsletter.typal.academy/p/proving-two-sets-are-equal</guid><dc:creator><![CDATA[Howard Heaton]]></dc:creator><pubDate>Wed, 16 Oct 2024 13:01:27 GMT</pubDate><enclosure url="https://substackcdn.com/image/fetch/$s_!EIkC!,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2Fc417dbe4-9cc5-488c-a492-b34aae0757c7_1920x2400.png" length="0" type="image/jpeg"/><content:encoded><![CDATA[
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   ]]></content:encoded></item><item><title><![CDATA[Limits at Infinity]]></title><description><![CDATA[Refresher on how to formalize limit arguments]]></description><link>https://newsletter.typal.academy/p/limits-at-infinity</link><guid isPermaLink="false">https://newsletter.typal.academy/p/limits-at-infinity</guid><dc:creator><![CDATA[Howard Heaton]]></dc:creator><pubDate>Wed, 09 Oct 2024 13:01:47 GMT</pubDate><enclosure url="https://substackcdn.com/image/fetch/$s_!Liz4!,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F0a2a1393-56f6-4a9b-b75b-993479065a93_1920x1439.png" length="0" type="image/jpeg"/><content:encoded><![CDATA[
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   ]]></content:encoded></item><item><title><![CDATA[Rolle's Theorem]]></title><description><![CDATA[Equal endpoints + differentiable = derivative is zero somewhere in between.]]></description><link>https://newsletter.typal.academy/p/rolles-theorem</link><guid isPermaLink="false">https://newsletter.typal.academy/p/rolles-theorem</guid><dc:creator><![CDATA[Howard Heaton]]></dc:creator><pubDate>Fri, 26 Jul 2024 14:02:20 GMT</pubDate><enclosure url="https://substackcdn.com/image/fetch/$s_!GKXs!,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2Fa2ee51fc-fe58-410d-b893-27f68d27b799_1920x1439.png" length="0" type="image/jpeg"/><content:encoded><![CDATA[
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   ]]></content:encoded></item><item><title><![CDATA[ε-δ Perspective on Derivatives]]></title><description><![CDATA[Establishing convergence of secant lines]]></description><link>https://newsletter.typal.academy/p/perspective-on-derivatives</link><guid isPermaLink="false">https://newsletter.typal.academy/p/perspective-on-derivatives</guid><dc:creator><![CDATA[Howard Heaton]]></dc:creator><pubDate>Fri, 12 Jul 2024 22:00:03 GMT</pubDate><enclosure url="https://substackcdn.com/image/fetch/$s_!C6Td!,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F6c8b73f3-d52b-43b8-949f-66747f693674_2400x1800.png" length="0" type="image/jpeg"/><content:encoded><![CDATA[
      <p>
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   ]]></content:encoded></item><item><title><![CDATA[Relating ε-δ Continuity to Open Inverses]]></title><description><![CDATA[A proof of one direction of the equivalence]]></description><link>https://newsletter.typal.academy/p/relating-continuity-to-open-inverses</link><guid isPermaLink="false">https://newsletter.typal.academy/p/relating-continuity-to-open-inverses</guid><dc:creator><![CDATA[Howard Heaton]]></dc:creator><pubDate>Fri, 05 Jul 2024 19:30:57 GMT</pubDate><enclosure url="https://substackcdn.com/image/fetch/$s_!NU7c!,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F121f6342-2a8c-44fc-803b-daaabf03272c_1441x1080.gif" length="0" type="image/jpeg"/><content:encoded><![CDATA[
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          <a href="https://newsletter.typal.academy/p/relating-continuity-to-open-inverses">
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   ]]></content:encoded></item><item><title><![CDATA[Positivity and Continuity]]></title><description><![CDATA[Picking &#949; in the definition of continuity]]></description><link>https://newsletter.typal.academy/p/positivity-and-continuity</link><guid isPermaLink="false">https://newsletter.typal.academy/p/positivity-and-continuity</guid><dc:creator><![CDATA[Howard Heaton]]></dc:creator><pubDate>Fri, 21 Jun 2024 18:00:58 GMT</pubDate><enclosure url="https://substackcdn.com/image/fetch/$s_!0TKW!,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F7a64ddcd-46fc-4700-8d55-44318400998d_2400x1350.png" length="0" type="image/jpeg"/><content:encoded><![CDATA[
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   ]]></content:encoded></item><item><title><![CDATA[Pointwise Convergence with Rounded Tents]]></title><description><![CDATA[Fixing x and then taking a limit over n.]]></description><link>https://newsletter.typal.academy/p/pointwise-convergence-with-rounded-tens</link><guid isPermaLink="false">https://newsletter.typal.academy/p/pointwise-convergence-with-rounded-tens</guid><dc:creator><![CDATA[Howard Heaton]]></dc:creator><pubDate>Fri, 14 Jun 2024 22:00:47 GMT</pubDate><enclosure url="https://substackcdn.com/image/fetch/$s_!nGHK!,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2Fc7aca081-a2f8-4869-bbb9-bcc6d56884c7_1080x810.gif" length="0" type="image/jpeg"/><content:encoded><![CDATA[
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   ]]></content:encoded></item><item><title><![CDATA[Composition of Continuous Functions in 2D]]></title><description><![CDATA[A visual proof sketch]]></description><link>https://newsletter.typal.academy/p/composition-of-continuous-functions</link><guid isPermaLink="false">https://newsletter.typal.academy/p/composition-of-continuous-functions</guid><dc:creator><![CDATA[Howard Heaton]]></dc:creator><pubDate>Fri, 07 Jun 2024 13:01:26 GMT</pubDate><enclosure url="https://substackcdn.com/image/fetch/$s_!AlX-!,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2Ff3bce09a-b4f4-45e4-9b8f-710dcbde0cd7_1441x1080.gif" length="0" type="image/jpeg"/><content:encoded><![CDATA[
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   ]]></content:encoded></item><item><title><![CDATA[Banach Fixed Point Theorem via Cosine]]></title><description><![CDATA[Iteratively solving x = cos(x)]]></description><link>https://newsletter.typal.academy/p/banach-fixed-point-theorem-via-cosine</link><guid isPermaLink="false">https://newsletter.typal.academy/p/banach-fixed-point-theorem-via-cosine</guid><dc:creator><![CDATA[Howard Heaton]]></dc:creator><pubDate>Wed, 29 May 2024 14:01:42 GMT</pubDate><enclosure url="https://substackcdn.com/image/fetch/$s_!JsTL!,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F78b097f4-5de2-4a15-9510-5ba66627b086_1200x1200.gif" length="0" type="image/jpeg"/><content:encoded><![CDATA[
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          <a href="https://newsletter.typal.academy/p/banach-fixed-point-theorem-via-cosine">
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   ]]></content:encoded></item><item><title><![CDATA[Uniformly Continuous Function]]></title><description><![CDATA[Same &#948; for all points in the domain]]></description><link>https://newsletter.typal.academy/p/uniformly-continuous-function</link><guid isPermaLink="false">https://newsletter.typal.academy/p/uniformly-continuous-function</guid><dc:creator><![CDATA[Howard Heaton]]></dc:creator><pubDate>Wed, 22 May 2024 15:00:00 GMT</pubDate><enclosure url="https://substackcdn.com/image/fetch/$s_!Qcfr!,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F9130d53e-57ef-42bc-acfd-e7b53e17a24f_1920x1440.gif" length="0" type="image/jpeg"/><content:encoded><![CDATA[
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   ]]></content:encoded></item><item><title><![CDATA[Proof for Limit of Quadratic Function]]></title><description><![CDATA[Picking &#948; as the minimum of two numbers]]></description><link>https://newsletter.typal.academy/p/proof-for-limit-of-quadratic-function</link><guid isPermaLink="false">https://newsletter.typal.academy/p/proof-for-limit-of-quadratic-function</guid><dc:creator><![CDATA[Howard Heaton]]></dc:creator><pubDate>Mon, 13 May 2024 15:00:00 GMT</pubDate><enclosure url="https://substackcdn.com/image/youtube/w_728,c_limit/uPH-1wIrKGM" length="0" type="image/jpeg"/><content:encoded><![CDATA[
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          <a href="https://newsletter.typal.academy/p/proof-for-limit-of-quadratic-function">
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   ]]></content:encoded></item><item><title><![CDATA[Alternating Harmonic Series]]></title><description><![CDATA[Insight into a peculiar series that approximates an integral]]></description><link>https://newsletter.typal.academy/p/alternating-harmonic-series</link><guid isPermaLink="false">https://newsletter.typal.academy/p/alternating-harmonic-series</guid><dc:creator><![CDATA[Howard Heaton]]></dc:creator><pubDate>Wed, 01 May 2024 15:00:00 GMT</pubDate><enclosure url="https://substackcdn.com/image/fetch/$s_!ZlED!,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F3bc74955-0455-477f-af21-f4c4e04baed8_800x800.gif" length="0" type="image/jpeg"/><content:encoded><![CDATA[
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   ]]></content:encoded></item><item><title><![CDATA[Bolzano-Weierstrass Theorem]]></title><description><![CDATA[Bounded sequences have convergent subsequences]]></description><link>https://newsletter.typal.academy/p/bolzano-weierstrass-theorem</link><guid isPermaLink="false">https://newsletter.typal.academy/p/bolzano-weierstrass-theorem</guid><dc:creator><![CDATA[Howard Heaton]]></dc:creator><pubDate>Sun, 21 Apr 2024 15:00:00 GMT</pubDate><enclosure url="https://substackcdn.com/image/fetch/$s_!AEYJ!,f_auto,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2Fc012a16f-02fa-4c0d-bb9d-2b38bfc0bb03_1920x1600.gif" length="0" type="image/jpeg"/><content:encoded><![CDATA[
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