Justifying that infinite sum of solutions to wave equation is also a solution

Question

I'm currently learning about the separation of variables technique for solving $u_{tt} = c^2 u_{xx}, \; 0 < x < l$ with $u(0,t) = 0 = u(l,t)$ (section 4.1 of Partial Differential Equations by Strauss). After showing that $$u_n(x,t) = \bigg(A_n \cos \frac{n \pi c t}{l} + B_n \sin \frac{n \pi ct}{l}\bigg) \sin \frac{n \pi x}{l}$$ is a solution for any $n \in \mathbb{Z}$ and arbitrary constants $A_n$ and $B_n$, the book goes on to give the general solution as an infinite sum: $$u(x,t) = \sum_{n=1}^{\infty} \bigg(A_n \cos \frac{n \pi c t}{l} + B_n \sin \frac{n \pi c t}{l}\bigg) \sin \frac{n \pi x}{l}.$$ My question is: what justifies asserting that the infinite sum is also a solution? I agree that any finite sum $\sum_{n=1}^N u_n(x,t)$ is also a solution by the superposition principle (a.k.a. linearity), but in the book this principle is stated specifically for finite sums. Could someone provide me some justification for this? I would like to see a rigorous argument if it's not too complicated, or at least the main ideas behind the argument. Thanks!