Sequences and closed sets, Real Analysis II

In this lecture, I link the concept of sequential convergence to accumulation points and closed sets within a metric space. I start by proving that a set A is closed if and only if every sequence in A that converges has its limit also within A. This establishes that in closed sets, limits of converging sequences cannot fall outside the set, unlike in non-closed sets.

(MA 426 Real Analysis II, Lecture 13)

To tackle this proof, we reframe the statement: A set is not closed if and only if there exists a sequence within the set that converges to a limit outside the set. I construct such a sequence by using an accumulation point in the complement of A and demonstrate how this sequence converges to the point not in A, showing the set is not closed.

I then discuss a related concept: how to define the closure of a set B. I prove that the closure of B can be characterized by the limits of sequences within B, reinforcing the idea that the closure is essentially the set along with its external accumulation points.

This lecture helps us understand topological properties like closed sets, setting the stage for further study, including compact sets.

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