give an example a of a sequence of functions with a certain property Give an example of a sequence of functions on $[0,1]$ with the property that $\lim_{ n→∞} f_n(x) = 0$ for all 0 ≤ x ≤ 1 and yet for every interval $[c,d] ⊂ [0,1]$ and every N there is some x ∈ [c,d] and n > N with $f_n(x) > 1$.
 A: Look at 1. from Saz's answer to this question: Convergence types in probability theory : Counterexamples.
The idea of this, and any other such counterexample, is that you can move around the points $y$ where $f_n(y) > 1$ as $n$ increases such that each point can have limit $0$ (because for any given point $x$ $f_n(x)>1$ only finitely many times) but no matter how large $n$ is there is always at least some point $z$ for which $f_n(z) >1$ even if for most points the opposite is true. Thus this possibility depends in large part on the fact that $[0,1]$ is an infinite set.
A: Take $\{q_n\}_{n\geq 0}$ be the set rational numbers in $[0,1]$ then let
$$f_n(x)=\begin{cases}
2\quad\mbox{if $x=q_m$ for $m\geq n$},\\
0\quad\mbox{otherwise.}\\
\end{cases}$$
Clearly for all $x\in [0,1]$, $\displaystyle \lim_{ n→∞} f_n(x) = 0$. Moreover,  for every interval $[c,d]\subset [0,1]$, since the sequence 
$\{q_n\}_{n\geq N}$ is dense for any $N$, there is a $x:=q_n\in [c,d]$ for $n>N$ and by definition $f_n(x)=2>1$.
