Let $A$ and $B$ be two bounded sets of real numbers. Prove that $A\cup B$ is bounded Quick question. What example can I consider to prove the following statement and how do you prove it in the simplest way possible ?

Let $A$ and $B$ be two bounded sets of real numbers. Prove that $A\cup B$ is bounded.

 A: For a set $A$ to be bounded, this means that there exists a positive integer $N$ such that $-N<a<N$ for all $a\in A$. Now, since $B$ is bounded, we have the existence of a positive integer $M$ such that $-M<b<M$ for all $b\in B$.
Define $Z=\max\{M,N\}$. I claim that $-Z< c<Z$ for all $c\in A\cup B$. Suppose not. Then there is some $c\in A\cup B$ such that $|c|\geq Z=\max\{M,N\}$. If $c\in A$, then we have $N>|c|\geq \max\{M,N\}$, a contradiction. Clearly we can apply the argument if $c\in B$ and get another contradiction. This means that $-Z<c<Z$ for all $c\in A\cup B$, satisfying the definition of bounded.
A: Why would you ask for an example?  An example can prove existence; it can't prove universal statements.
Suppose every member of $A$ is within a certain distance $r_1$ of some particular point $x_1$, and every member of $B$ is within a certain distance $r_2$ of some particular point $x_2$.  Suppose the distance from $x_1$ to $x_2$ is $d$.
Then every member of $A\cup B$ is within $\max\{r_1, d+r_2\}$ of $x_1$.  That can be shown using the triangle inequality for $d+r_2$.
A: A set in the real line is bounded iff it is contained in a closed interval of finite length. Given two such intervals, it is easy to find another such interval containing them; this interval bounds the original two sets.
A: My definition of "bounded" is this:
A set $S$ of real numbers is bounded (above, below) if
there are real numbers $p$ and $q$ (real number $p$) such that,
for all $s \in S$, $q \le s \le p$
($s \le p$, $s \ge p$).
If $A$ has lower bound $a1$ and upper bound  $a2$ and similarly for $B$,
then $\min(a1, b1)$ and $\max(a2, b2)$ are,
respectively,
lower and upper bounds for $A \cup B$.
The proof follows from the definitions of min, max, and union.
OK - what am I overlooking?
