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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/3CMdgjI We prove that if a monotone sequence has a convergent subsequence, as in A SINGLE CONVERGENT SUBSEQUENCE, then the monotone sequence converges and has the same limit as its subsequence. Of course, this means that in fact all of its subsequences converge to the same thing, but the point is we only need to know of one convergent subsequence, and then we can conclude a monotone sequence converges too. #RealAnalysis For this proof, we'll need our subsequence limit theorem: stating a sequence converges if and only if all of its subsequences converge to the same thing. We prove that here: https://www.youtube.com/watch?v=0oRN_pxq2IM We'll also need the monotone convergence theorem, which tells us how monotone sequences behave when they are bounded or convergent. We prove it here: https://www.youtube.com/watch?v=4NURDmE79VU Two other small facts we use are 1) if we go k terms into a subsequence, then we are at least k terms into the original sequence. As in: n_k \geq k, which we talk about here: https://www.youtube.com/watch?v=5ZSzzVd9X6U and 2) if a sequence is monotone then so are all of its subsequences, which we prove here: https://youtu.be/pPFLXA4Mzpc We'll prove the result only for increasing sequences, the decreasing case is very similar, try proving it yourself! Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath ● Twitter: https://twitter.com/wrathofmathedu