In the lecture I mentioned that there is a gap in the proof that closed bounded subsets of R are sequencially compact. The gap was, what happens if the l.u.b C = b?

If b is in C, then only finitely many members of the sequence are less than b. Hence, b must be in the sequence infinitely often. Then we can choose a subsequence of all b's which will converge to b. In fact, the original sequence would converge to b.

Suppose b is not in C. But then there are only a finite number of terms in the sequence. This contradicts that we are working with an infinite sequence and completes the proof.