Question

Let \(K\) be a compact subset of \(\mathbb{R}\) and let \(p \in\ \mathbb{R} \) \ \(K\). Prove there exists \(q \in\ K\) such that \(|q-p| =\)\(inf\) {\(|x-p| | x \in\ K\)}

Answer 1

@ganeshie8 @eliassaab

Answer 2

@mathmath333 Any Idea?

Answer 3

@myininaya @aaronq

Answer 4

@wio Help,Like you're the only one left,lmao

Answer 5

@SithsAndGiggles

Answer 6

@Alchemista is our best man.

Answer 7

@ikram002p @Loser66

Answer 8

Not sure if I have the right idea here, and certainly not a formal proof, but maybe you'll find it useful. Since \(K\) is compact, you're essentially considering a closed and bounded (see Heine-Borel theorem) interval on the real line.

I think the fact that it's closed/bounded should allow you to conclude that the endpoint closest to \(p\) is precisely the \(q\) that satisfies the problem statement.