I am reading the Wikipedia article on the solution of the one dimensional heat equation but I don't understand one of the steps.
In (7) they write that $X(0) = 0 = X(L)$ follows from $u\mid_{x=0} = u\mid_{x=L}=0$ for $X(x) = b e^{\sqrt{-\lambda} x} + c e^{-\sqrt{-\lambda} x}$.
Why does this follow?
We have $u(x,t) = X(x) T(t)$ so that $u(x=0, t) = X(0) T(t)$ and $u(x=L,t) = X(L) T(L)$.
With $X(x) = b e^{\sqrt{-\lambda} x} + c e^{-\sqrt{-\lambda} x}$ and therefore $X(0) = b+c$ we get $X(0) T(t) = (b+c)T(t)$.
When separating variables, we're looking for solutions that are not identically zero, so we can assume that there is some $t_0$ such that $T(t_0) \neq 0$. Since the boundary conditions must hold for all $t$, they hold for $t=t_0$ in particular, and thus $$ 0 = u(0,t_0) = X(0) \, T(t_0) , $$ which implies that $X(0)=0$, and likewise $$ 0 = u(L,t_0) = X(L) \, T(t_0) , $$ which implies that $X(L)=0$.
(The fact that $X(x)$ is a sum of exponentials is irrelevant for this argument.)