Sequentially compact sets and totally bounded sets, Real Analysis II

In this video, I explain sequentially compact sets and total boundedness in a metric space. A set is sequentially compact if every sequence in the set has a subsequence that converges to a point within the set. We illustrate this with examples such as open and closed intervals, the Archimedean set, integers, and subsets of integers and rational numbers, highlighting certain properties and strategies to look for.

(MA 426 Real Analysis II, Lecture 15)

Then we discuss total boundedness, a stronger form of boundedness, and prove that every totally bounded set is bounded. Lastly we prove that sequentially compact sets are totally bounded by proving the contrapositive. This lecture sets the stage for the next lesson on compact sets.

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