Consider the next set.

$\displaystyle \Lambda = \left\{ {f \in C\left( {\left[ {0,1} \right],\mathbb{R}} \right);\left| {f\left( x \right) - f\left( y \right)} \right| \leqslant \pi \left| {x - y} \right| \wedge \left\| f \right\|_\infty \leqslant 1} \right\}$

I have been ask to prove that $\displaystyle \Lambda in \left( {C\left( {\left[ {0,1} \right],\mathbb{R}} \right),\left\| . \right\|_\infty } \right)\$ is compact.

I really don`t know how to do it. I have no idea where to start from. Please help.

Regards.