i have this question about series. So i came along a question that started with : $(f_{n})_{n}$:[-1,1]->$R^{+}$ is a row of continuous functions. The serie $\sum{f_{k}(x)}$ from k=0 to $\inf$ will converges and thefunction f:[-1,1]->$R^{+}$:x->$\sum(f_{k}(x))$ is continuous.
Now i wanted to prove that $\sum f_{k}$ uniformly converges.
I wanted to use the M-test from weierstrass. Therefore i only needed to prove that |$f_{k}(x)|$<= $f_{k}(x)$. For this i wanted to use that f is continous but i'm stuck and not sure if we can even say that it uniformly converge.
Tha follows from Dini's theorem: since each $f_n$ take only non-negative values, the sequence $\left(\sum_{n=1}^Nf_n(x)\right)_{N\in\Bbb N}$ is monotonic increasing for each $x\in[-1,1]$. So, since each $\sum_{n=1}^Nf_n$ is continuous and since $f$ is continuous, the convergence is uniform.