I have been reading An Introduction to PDE written by Pinchover and Rubinstein, but there is a proof of a corollary that is not clear for me.
Once they proof weak maximum principle for heat equation and use it to prove continuous dependence of solutions with parameters and uniqueness a corollary says the following
Let \begin{align*} u(t,x)=\sum_{n=1}^{\infty}B_{n}\sin\frac{n\pi x}{L}e^{-k\left(\frac{n\pi}{L}\right)^{2}t} \end{align*} be a formal solution of the heat problem \begin{align*} \begin{cases} u_{t}-ku_{xx}=0 &x\in(0,L),\,t>0\\ \begin{cases} u(t,0)=0 &t\geq0\\ u(t,L)=0 &\\ u(0,x)=f(x) &x\in[0,L]\\ \end{cases} \end{cases} \end{align*}
If series \begin{align*} f(x)=\sum_{n=1}^{\infty}B_{n}\sin\frac{n\pi x}{L} \end{align*} converges uniformly in $[0,L]$, then the series for $u$ converges uniformly in $[0,L]\times[0,T]$, and $u$ is classical.
For the proof uses the Cauchy criterion for uniform convergence for the series of $f(x)$, that is, for $\epsilon>0$ exists $N_{\epsilon}$ such that for $l\geq k\geq N_{\epsilon}$ is true that \begin{align*} \left\vert S_{l}-S_{k}\right\vert<\epsilon,\,\forall x\in[0,L] \end{align*} where I mean for $S_{k}$ \begin{align*} S_{k}=\sum_{n=1}^{k}B_{n}\sin\frac{n\pi x}{L} \end{align*} that is \begin{align*} \left\vert \sum_{n=1}^{l}B_{n}\sin\frac{n\pi x}{L}-\sum_{n=1}^{k}B_{n}\sin\frac{n\pi x}{L}\right\vert< &\epsilon,\,\forall x\in[0,L]\\ \left\vert \sum_{n=k}^{l}B_{n}\sin\frac{n\pi x}{L}\right\vert<& \end{align*}
Obviously, for each $n$ we have that $u_{n}(t,x)=B_{n}\sin\frac{n\pi x}{L}e^{-k\left(\frac{n\pi}{L}\right)^{2}t}$ is a classical solution of the heat equation and, by superposition principle, a finite sum of them are again classical solutions, that is \begin{align*} v(t,x)=\sum_{n=k}^{l}B_{n}\sin\frac{n\pi x}{L}e^{-k\left(\frac{n\pi}{L}\right)^{2}t} \end{align*} is classical. Here, the autors say that, by the weak maximum principle, we have \begin{align*} \left\vert\sum_{n=k}^{l}B_{n}\sin\frac{n\pi x}{L}e^{-k\left(\frac{n\pi}{L}\right)^{2}t}\right\vert<\epsilon,\,\forall x\in[0,L] \end{align*} Then they complete the proof saying that by Cauchy criterion the series solution converges uniformly on the square to a continuous function $u$ that satisfies boundary and initial conditions, therefore $u$ is classical.
WHAT I DON'T UNDERSTAND IS THE JUMP USING THE MAXIMUM PRINCIPLE. CAN ANYONE EXPLAIN IT TO ME IN A CLEAR AN EXPLICIT WAY?