MATH 242: Analysis 1 (McGill Final)

Currently Playing: Proof: Cauchy Sequences are Convergent | Real Analysis

Support the production of this course by joining Wrath of Math to access all my real analysis videos plus the lecture notes at the premium tier! https://www.youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join 🛍 Get the coolest math clothes in the world! https://mathshion.com/ Real Analysis course: https://www.youtube.com/playlist?list=PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Real Analysis exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxXAN05Gm3iNmpz9SkVfLNqC Get the textbook! https://amzn.to/45kcMjq We prove every Cauchy sequence converges. To do this we use the fact that Cauchy sequences are bounded, then apply the Bolzano Weierstrass theorem to get a convergent subsequence, then we use Cauchy and subsequence properties to prove the sequence converges to that same limit as the subsequence. With our previous proofs, we will have now proven a sequence converges if and only if it is Cauchy. #realanalysis Proof Sequence Converges if and Only if all of its Subsequences Do: https://youtu.be/0oRN_pxq2IM Proof of Bolzano-Weierstrass Theorem (coming soon): Intro to Cauchy Sequences: https://youtu.be/VNoHcFoawTg Proof Cauchy Sequences are Bounded: https://youtu.be/GulH7nS_65c Proof Every Convergent Sequence is Cauchy: https://youtu.be/SubZMuVBajM ★DONATE★ ◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: https://www.patreon.com/join/wrathofmathlessons ◆ Donate on PayPal: https://www.paypal.me/wrathofmath Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath ● Twitter: https://twitter.com/wrathofmathedu