MATH 242: Analysis 1 (McGill Final)

Currently Playing: Proof: Sequence is Cauchy if and only if it Converges | 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 that a sequence converges if and only if it is Cauchy! This means that if a sequence converges then it is Cauchy, and if a sequence is Cauchy then it converges. Note that the definition of a Cauchy sequence has nothing to do with the particular limit of the sequence, so this gives us a way to prove a sequence converges without knowing its limit - we simply prove it is Cauchy! Before we prove either direction of this result we will prove that Cauchy sequences are bounded, since we use this result in the proof. With this proof done, we have got an awful lot of value from the Cauchy criterion for sequences! #RealAnalysis 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 Proof Sequence Converges if and Only if all of its Subsequences Do: https://youtu.be/0oRN_pxq2IM Proof of Bolzano-Weierstrass Theorem (coming soon): Proof Cauchy Sequences Converge: https://youtu.be/xhBfPoSjAR0 Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath ● Twitter: https://twitter.com/wrathofmathedu 0:00 Intro 2:14 Every Cauchy Sequence is Bounded 7:17 Every Convergent Sequence is Cauchy 12:31 Every Cauchy Sequence is Convergent