Question

Let A be a nonempty compact subset of R and c in R. Prove that there exists a point a in A such that |c-a|=inf{|c-x|:x∈A}.

Answer 1

This seems somewhat obvious to me. Since \(x\in A\), and we want to choose some \(a\in A\) for a determined \(c\), let \(B=\{|c-x| : x\in A\}\). Now look at the set \(C=\{|c-a| : a \in A\}\). This is the same set. So the infimum of \(B\) must be the same as the infimum of \(C\). So we are done.

Answer 2

Let d=that infimumu. For every n>1, there is
\[ x_n \in A\] such that
\[| c- x_n| \le d + \frac 1 n
\]
Since A is compact then there is a subsequnce
\[ x_{n_k}
\]
that converges to an elemnt a in A.
Hence
\[
d \le | c- x_{n_k}| \le d +\frac 1{n_k}
\]
when
\[
n_k \to \infty , \text { we have } \\
d \le | c- a| \le d

\]
You are done