Prove $f(x) = \sum^\infty_{k=1}A_k \sin(k\pi x) = 0 \Rightarrow A_k = 0, \forall k \in \mathbb{N}_+ $

Question

I'm trying to solve 2(c) of this exam.

How can you constructively prove:

In $\LaTeX$: $$f(x) = \sum^\infty_{k=1}A_k \sin(k\pi x) = 0 \Rightarrow A_k = 0, \forall k \in \mathbb{N}_+ $$

In my head, what if $x = 0 $, then $f(0) = 0$, and therefore $A_k$ can be anything.