PROOF A Convergent Sequence is ALWAYS Bounded! | Real Analysis Lecture!

We rigorously prove that a convergent sequence in Euclidean space R^n is always bounded. We review the rigorous definition of a bounded sequence in R^n and the rigorous ``epsilon-delta" definition of a limit (the rigorous definition of convergence) and visualize it in terms of epsilon-balls (and neighborhoods) in R^n. We explain the intuition behind this seemingly obvious fact, and how to formalize the intuition and write a rigorous math proof in order to practice mastery of real analysis.

The video utilizes the standard metric on Euclidean space, and the fundamental properties it satisfies, such as the triangle inequality. The video is designed for people who wish to master the theory of sequences, and who are interested in learning rigorous proof-based math, and introductory real analysis, from fundamentals! 👑

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