- #1

Bashyboy

- 1,421

- 5

## Homework Statement

I am trying to understand the proof that ##\lim S## is a closed set in the metric space ##M##, where ##\lim S = \{ p \in M ~|~ p \mbox{ is a limit point of } S\}##.

Here is the definition of a limit point: ##p## is a limit point of ##S## if and only if there exists a sequence ##(p_n)## of points in ##S## such that ##p_n \rightarrow p##. Attached is a snapshot from the book.

## Homework Equations

## The Attempt at a Solution

I don't understand why there exists ##q_n = p_{n,k(n)}## satisfying the conditions mentioned in the picture. What does ##k(n)## denote? A subsequence? The indices are causing me trouble. Why would only one of these sequences converge to ##p##? It seems that all of them should. Clearly ##p_{n,k} \rightarrow p_n## means ##d(p_n,p_{n,k}) < \frac{1}{k}##, but I don't see the connection to what is give in the picture. How does the replacement of ##k## with ##k(n)## factor into the proof? I tried showing there exists such a ##q_n##, but was unsuccessful.