Limit of supremum norm of a series of functions that converge pointwise

Question

Suppose $\{f_n\}_{i = 1}^\infty$ is a sequence of bounded functions $[0; 1] \to \mathbb{R}$, that converges uniformly to $f$. Does that imply that $\lim_{n \to \infty} \left\|f_n\right\|_{\infty} = \left\|f\right\|_{\infty}$?

I think that it has something to do, that the uniform convergence is exactly the convergence in supremum norm. However, I do not know how to apply this fact in order to prove this statement...

Answer 1

$ |\|f_n\|-\|f\| |\leq \|f_n-f\| \to 0$ by the reverse triangle inequality so $ \|f_n\|\to \|f\|$