- #1

- 317

- 26

## Homework Statement

Let ##K\neq\emptyset## be a compact set in ##\Bbb{R}## and let ##c\in\Bbb{R}##. Then ##\exists a\in K## such that ##\vert c-a\vert=\inf\{\vert c-x\vert : x\in K\}##.

**2. Relevant results**

Any set ##K## is compact in ##\Bbb{R}## if and only if every sequence in ##K## has a subsequence that converges to a point in ##K##.

## The Attempt at a Solution

I was hinted that ##\inf\{|c-x|\mid x\in K\} \leq |c-x_n|\lt \inf\{|c-x|\mid x\in K\}+\frac{1}{n}##. Now it became clearer that if I was to set ##\vert c-a\vert=\inf\{|c-x|\mid x\in K\}##, then that means there exist as sequence ##(x_n)## in ##K## such that it converges to ##a\in\Bbb{R}##. Since ##K## is compact then it must contain ##a##. However, I think this argument is too informal. Please help me improve on it. Thank you.