I need to show that for $K>0$, $$X=\{f:[0,1]\rightarrow [0,1]\mid |f(x)-f(y)|\leq K|x-y|\ \forall x,y \in [0,1]\}$$ with the metric $d(f,g)=\max|f(x)-g(x)|$ , (supremum metric), is a compact space. Using the lemma: $X$ metric and complete is compact if and only if for any $A$ infinite subset of $X$ and for any $\epsilon>0$ $\exists x,y \in A$ s.t. $d(x,y)<\epsilon$.
I've proved the lemma and that the given $(X,d)$ is complete space.
Thank you!
I like the following "graphical" approach. Given $\epsilon>0$, choose $m>2/\epsilon$ and $n>Km$; then divide the intervals $[0,1]$ on the vertical and horizontal axes into $m$ and $n$ equal subintervals, respectively. We now have a rectangular grid consisting of $mn$ rectangles. The set of all rectangles that intersect the graph of $f$ is the "signature" of $f$. Observe that
There are only finitely many possible signatures.
If two functions $f,g$ have the same signature, then $d(f,g)<\epsilon$.
To prove item 2, it helps to show first that in every column of the grid, the signature can contain at most two rectangles, and those will be adjacent.
Any infinite subset $A$ will contain two functions with the same signature, which yields the result.
You can translate all of this into interval notation, without referring to any grids. I just find the graphical interpretation helpful.