TITLE: Pointwise convergence implies $L^{2}$ convergence
QUESTION [6 upvotes]: If we have  a sequence of bounded functions $f_{n}$ converging almost everywhere to another bounded function $f$ in a finite measure space such that
$$ |f_{n}(t)| \leq c$$ for some constant $c$. Then
$$ \int |f_{n}(t) - f(t)|^{2} d\mu$$ goes to zero.
Since we are in a finite measure space, the constants are integrable and hence DCT applies. However, it only gives the result that
$$ \int f_{n}(t)d\mu \rightarrow \int f(t)d\mu$$ and
$$ \int |f_{n}(t) - f(t)| d\mu \rightarrow 0$$.
How do we prove the convergence in $L^{2}$ norm?
Here is what I tried:
$$ \int |f_{n}(t) - f(t)|^{2} d\mu = \int (f_{n}-f)(\overline{f_{n}}-\overline{f})$$
$$ = \int |f_{n}|^{2} - \int f\overline{f_{n}} - \int f_{n}\overline{f} + \int |f|^{2} $$
which goes to zero by DCT (as $f_{n}\overline{f} \rightarrow |f|^{2}$ and so on).
Am I right in all this or am I missing something? I am apprehensive because I intuitively feel that pointwise convergence should not imply $L^{2}$ convergence.

REPLY [13 votes]: You are right, but you can also argue more directly. Let $g_n(t) = |f_n(t) - f(t)|^2$. Then $g_n \to 0$ almost everywhere and $|g_n| \le 4c^2$ almost everywhere. As we are in a finite measure space, 
$$ \|f_n - f\|_{L^2}^2 =  \int g_n  \, d\mu \to 0. $$