Bounded and finite integral If $f:\mathbb R \rightarrow [0,\infty]$ is bounded and $\int f<\infty$, then $\int f^2<\infty$.
I'm trying to write $f$ in terms of a finite linear combination of characteristic functions, $f=\sum^n_{k=0} a_k.\chi_{E_k}$ with ${E_k}$'s disjoint.
Then I get that $f^2=\sum^n_{k=0} a^2_k.\chi_{E_k}$ (Is this right?).
How else can I proceed from here?
 A: $$
0 \le f \le M \implies \int f^2  \le \int M f
$$
So...
A: Say $f\geq 0$ is bounded above by $M$. Then $\int f^2\leq \int M f= M\int f<\infty$.
A: I've seen that some people have already given some very concise answers. Here is a technique that can be used to deal with a variety of $L^p$-related problems, not just this one. Basically, $L^p$ functions can fail to be integrable because of two possible things: either (1) they have a singularity that grows too fast, or (2) they don't remain small enough out at infinity. 
This motivates decomposing a positive $f$ into $f = f_1 + f_2$, where $f_1 = f$ whenever $f \geq 1$, and $f_1 = 0$ otherwise, and $f_2 = f$ whenever $f < 1$, and $f_2 = 0$ otherwise. Thus $f_1$ captures the "blow up" properties and $f_2$ captures the "decay" properties.
Then, to resolve your specific question, then $f^2 = f_1^2 + f_2^2$, and you can analyze each piece and how it behaves. But the point is that you can use this decomposition to relate $L^p$ and $L^q$ for different $p,q$, in a variety of common exercises.
A: Note that, $f$ is Lebesgue integrable iff $|f|$ is. So , we have
$$ \Big|\int f^2 \Big|  \leq  \int |f||f|  \leq M \int |f| < \infty .  $$
