Why does $L^2$ convergence not imply almost sure convergence
Consider the following sequence of characteristic functions $f_n \colon [0,1] \to R$ defined as follows:
$f_1 = \chi[0, 1/2]$
$f_2 = \chi[1/2, 1]$
$f_3 = \chi[0, 1/3]$
$f_4 = \chi[1/3, 2/3]$
$f_5 = \chi[2/3, 1]$
$f_6 = \chi[0, 1/4]$
$f_7 = \chi[1/4, 2/4]$
and so on.
Then $f_n \to 0$ in $L^2$, but $f_n$ does not converge pointwise.