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- #1

- Mar 10, 2012

- 835

Here'a a question I found in a Problem book:

Let $Y$ be a non-empty closed subset of a metric space $M$ and $x$ be any point in $M$. Show that $\inf\{d(x,y):y\in Y\}=d(x,y_0)$ for some $y_0\in Y$.

My approach is:

Say $I=\inf\{d(x,y):y\in Y\}$. Assume that the proposition is false. Now there exists a sequence $\{y_n\}$ in $Y$ such that $d(x,y_n)<I+1/n$ for all positive integers $n$. If I could somehow show that some subsequence of $y_n$ converges then I'd be done. I can do that if it were given that $M$ is compact but I am not able to prove without this extra hypothesis. Can anybody help?