Proof: Sequence is Cauchy if and only if it Converges | Real Analysis

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! Table of Contents 0:00 Intro 2:14 Every Cauchy Sequence is Bounded 7:17 Every Convergent Sequence is Cauchy 12:31 Every Cauchy Sequence is Convergent #RealAnalysis #Math 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 ★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 Thanks to Robert Rennie, Barbara Sharrock, and Rolf Waefler for their generous support on Patreon! Thanks to Crayon Angel, my favorite musician in the world, who upon my request gave me permission to use his music in my math lessons: https://crayonangel.bandcamp.com/ Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath ● Twitter: https://twitter.com/wrathofmathedu My Music Channel: https://www.youtube.com/channel/UCOvWZ_dg_ztMt3C7Qx3NKOQ