Trying to show that the infinite sum of $f_n(x)$ converges uniformly on $[0,1]$.
For $n=1,2,\ldots$ define continuous functions on $[0,1]$ by
- $f_n(x)=0$ on $\left[0,\frac{1}{2n+1}\right]$ or $\left[\frac{1}{2n-1},1\right]$,
- $f_n(x)=\frac{1}{n}$ if $x=\frac{1}{2n}$,
- $f_n(x)$ is linear on $[\frac{1}{2n+1},\frac{1}{2n}]$ and $\left[\frac{1}{2n},\frac{1}{2n-1}\right]$.
I sketched some graphs. I can see that when I plot a few of them the linear portions never overlap and that the peaks continue to decrease, however isn't the $1/n$ divergent by the Harmonic Series Theorem?