How to prove that a sequence of polygonal functions converges uniformly?

Question

Suppose that $\{\varphi_{n}\}$ is a sequence of polygonal functions from [0,1] to $X$, where $X$ is a compact set. How to prove that $\{\varphi_{n}\}$ converges uniformly to a certain $f$, or $\{\varphi_{n}\}$ is a Cauchy sequence? Any idea?

Answer 1

Let $X=\{0,1\}.$ For $x\in [0,1]$ let $\phi_n(x)=0$ if $n$ is even, and $\phi_n(x)=1$ if $n$ is odd..... No convergence..... A better Q would be whether (for any compact $X$) if $(\phi_n)_n$ has a point-wise or uniformly convergent subsequence.