Question

Let A be a non-empty 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 in A}.

Answer 1

A is a compact subset of R, so A is closed and bounded. We have
With \[c \in R, d(c,A)=\inf \left\{ d(c,x) : x \in A \right\} \]
Take y is a boundary point of A such that \[d(c,x)=d(c,y)+d(y,x)\]. Hence \[d(c,y)=d(c,x)-d(y,x)\le d(c,x),\forall x \in A\]
And if there exists b in R such that \[d(c,z)\ge b, \forall z \in A\], then \[b\le d(c,y)+d(y,z),\forall z \in A\]. In this inequality, we choose z=y, we have \[b\le d(c,y)\]
So \[d(c,y)=\inf \left\{ d(c,x):x \in A \right\}\]

Answer 2

"With c∈R,d(c,A)=inf{d(c,x):x∈A}" : remove this in my post

Answer 3

can you explain the last 2 step..from \[b \le d(c,y)\].....

Answer 4

I use the definition of inf