Simple uniform convergence quesiton

Question

Suppose f converges to f uniformly on $[0,L]$. Suppose |$f_{n}(x)$| ≤ $h(x)$ on $[0,\infty]$. Show |$f(x)$|≤ $h(x)$ on $[0,\infty]$.

This is probably really simple, I just can't wrap my head around why. Just wondered if someone could give me a simple explanation. Thank you.

Answer 1

Just take limit on both sides as $n \to \infty$. Pointwise convergence is enough for this.