Sums and products of bounded functions I have been on this one for hours, cant figure out how to write this in the proper format/wording.
Let $f$ and $g$ be functions from $\mathbb{R}$ to $\mathbb{R}$. For the sum and product of $f$ and $g$, determine which statements are true. If true provide a proof; if false, provide a counterexample.


*

*If $f$ and $g$ are bounded, then $f + g$ is bounded.

*If $f$ and $g$ are bounded, then $fg$ is bounded.

*If both $f + g$ and $fg$ are bounded, then $f$ and $g$ are bounded.


Any help is appreciated, thanks :)
 A: a) That $f$ is bounded means that $|f(x)| \leq C$ for all $x$. Similarly $|g(x)| \leq D$ for all $x$. Therefore the triangle inequality gives $|f(x) + g(x)| \leq |f(x)| + |g(x)| \leq C + D$ for all $x$ and hence $f+g$ is bounded by $C + D$.
b) is similar but easier
c) $f^2 + g^2 = |f^2 + g^2| = |(f + g)^{2} - 2fg| \leq |f + g|^2 + 2|fg|$ and the last term is bounded by assumption and a) and b).
A: Remember the definition of bounded: A function $h\colon\mathbb{R}\to\mathbb{R}$ is bounded if and only if there exist number $M,N\gt 0$ such that for all $r\in\mathbb{R}$, $-N \leq f(r) \leq M$.
Intuitively, the graph of $f$ goes neither "too high" nor "too low".
So, suppose that $f$ and $g$ are both bounded. That means that there the values of $f$ don't get bigger than some $M_1\gt 0$, and the values of $g$ don't get bigger than some $M_2\gt 0$. When you take $(f+g)(r) = f(r)+g(r)$, how big can the number $f(r)+g(r)$ be, given that $f(r)\leq M_1$ and $g(r)\leq M_2$?
How about discussing how small they can be, given that you know there are numbers $N_1$ and $N_2$ such that $f(r)\geq N_1$ and $g(r)\geq N_2$?
How about the product? That's a bit harder because of the signs, so try working with $|fg|$, $|f|$, and $|g|$. Notice that if $-N\leq f(r)\leq M$, then $0\leq f(r) \leq \max\{M,N\}$ (verify this!).
Number $3$ is a bit trickier... first question: can the sum of two functions, each of which is not bounded, be bounded? Yes: consider for instance $f(x) = x$, which is not bounded, and $g(x) = 1-x$, which is also not bounded. What is $f+g$?  Of course, this example does not satisfy that $fg$ is bounded (here, $fg(x) = x-x^2$, which is unbounded). And it's easy to come up with unbounded examples with their product bounded (just make sure that whenever $f$ gets out of control, $g$ "cancels it out"). 
Perhaps you may want to check that if $ff$ (the product of $f$ with itself) is bounded, then so is $f$, and then consider Theo Buehler's answer.
A: For a and b, you have that $f(x)\le f_{max}$ and similarly for g.  What does that tell you about $f(x)+g(x)$ and $f(x)g(x)$?  c is harder-it is easy for one of the conditions like $f(x)=-g(x)$ for the sum and $f(x)=0$ for the product, but for both together?
