Question

Let K be a non-empty compact subset of R and c in R. Prove that there exist a point "a" in K such that |c-"a"| = inf { |c-"a"| : x in K}

Answer 1

you mean inf of the distance between 2 sets: {a} and R?
And what is set A. Becouse you start saying K is subset of R and later A apears....

Answer 2

I'm sorry. I edited the question. Sorry for my typo :(

Answer 3

proof would go like this:

construct the function f(c) = |c-"a"|
then there are two posibilities:
1º if c=a it's trivial
2º if c distinct of a, the function is continuous on K and since K is compact it achieves there it's minimum and maximum.
That's it

Answer 4

I found hint but not for infimum but for supremum.

It goes like this,

If n in N (natural number), there exists \[a_{n} \in K\] such that \[\sup {|c-a|: a \in K } - 1/n < |c-a_{n}|\]
and asks to apply Bolzano-Weierstrass Theorem.

Answer 5

i gave you the proof already.