On a counter-example on the uniform convergence Let $f_n$ be a series of continuous function on $[a,b]$, such that $$\sum_{n=1}^\infty f_n(x)$$ converges for all $x\in [a,b]$. Moreover, $$f(x)=\sum_{n=1}^\infty f_n(x),x\in [a,b]$$ is continuous. Can we show that the series above is uniformly convergent on $[a,b]$?
I do think this was wrong, but I could not construct a counterexample...
 A: If for all $n=1,2,\cdots$, $f_n$ is no non-negative(meaning  for every $x\in [a,b],f_n(x)\geq 0$),then $\sum_{n=1}^\infty f_n$ converges uniformly to $f$ on $[a,b]$.See Dini theorem
A: My example is in the spirit of Petite Etincelle's comment (if they post an answer, I will delete), and illustrates the necessity of the "monotonically decreasing" hypothesis in Dini's Theorem. 
For $0\le x\le 1$, let $f_n(x)$ be given by the continuous, piecewise linear "tent" function

Then $f_n(x)\to f(x)\equiv 0$ pointwise (which is obviously a continuous function) on $[0,1]$, but the convergence is not uniform since $$\sup_{x\in[0,1]}|f_n(x)-0|=1\not\to 0\text{ as }n\to\infty.$$
Note that the requirement 
$$
f_{n+1}(x)\le f_n(x)\text{ for all }x\in[0,1] \text{ and all }n\in\mathbb N
$$
in Dini's Theorem fails here.

As side note, you can create examples to show that in fact all three of the major hypotheses in Dini's Theorem are necessary: 


*

*compactness of the set $I$ over which the $f_n, f$ are taken,

*continuity of the limit function $f$, 

*$f_{n+1}(x)\le f_n(x)$ for all $x\in I$ and all $n$.
