Metric space. Sequence convergence
Sequence convergence, definition
Let
[unfolding the definition of limits, this becomes...]
This next result should be internalized.
Lemma. Closed iff [some property of sequences...]
Let
Then
The proof was not immediately obvious to me. I did, however, find a way to visualize it, which I think makes the line of reasoning clear. Part of why the proof is important, I think, is that there are only two notions available from which the result needs to be derived: the definition of open sets and the definition of sequence convergence. So, it should be simple, yet it escapes an immediately obvious proof. Thus, I feel there is a mode of thinking that has its essence somewhat distilled into the line of reasoning.