I came across the proof of uniqueness of heat equation solution. Theorem: Let $u_1(x), u_2(x)$ be two solutions of the following: $$u_t= \alpha u_{xx}, 0<x<L, t>0$$ $$u(0,t)=g(t), u(L,t)=h(t)$$ $$u(x,0) = f(x)$$ where $f(x),g(t),h(t)$ are given functions. Then $u_1(x,t)=u_2(x,t)$ for all $0 \leq x \leq L, t>0$.
Proof: Set $w(x,t)=u_1(x,t) - u_2(x,t)$. Then $w$ satisfies $$w_t= \alpha w_{xx}, 0<x<L, t>0$$ $$w(0,t)=0, w(L,t)=0$$ $$w(x,0) = 0$$ Then, I do not understand the following part: By the maximum principle, we must have $$w(x,t) \leq 0$$
How does the maximum principle implies $w(x,t) \leq 0$ ? Please help. Thank you so much.