Proof: Cauchy Sequences are Bounded | 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 every Cauchy sequence is bounded. We previously discussed what the Cauchy criterion and Cauchy sequences are, and proved that a sequence is Cauchy. We are on our way to proving a sequence converges if and only if it is Cauchy, but to help us in doing that, we will first prove that Cauchy sequences are bounded in today's #realanalysis video lesson. To do this, we use the definition of Cauchy sequences to then use our useful absolute value inequality equivalence to establish a bound on terms after a certain point. Then, a max and min will clean up the rest. What are Bounded Sequences: https://www.youtube.com/watch?v=hkCKQdAFLAw Proof Convergent Sequence is Bounded: https://www.youtube.com/watch?v=7N-4XZlxBho Proof of the Absolute Value Inequality Equivalence: https://www.youtube.com/watch?v=X7GhczgUy7c Intro to Cauchy Sequences: https://youtu.be/VNoHcFoawTg Proof Convergent Sequences are 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