Analysis- proving that a sequence is bounded

Suppose that X,Y⊂ $\mathbb{R}$ are both bounded, prove that X $\cup$ Y is bounded.
I'm not sure how to go about doing the proof.

What's your definition of bounded?

A sequence ($\mathrm{x}_n$) is bounded if $\exists$ U $\in$ $\mathbb{R}$ such that |$\mathrm{x}_n$|≤U $\forall$n$\in$$\mathbb{N}$

Yes, this defines whether a sequence is bounded.

Unless you are using very strange nomenclature, the sets X, Y in your original question are not sequences, so you still haven't given us a definition of bounded that applies to X and Y.

Yes, this defines whether a sequence is bounded.

Unless you are using very strange nomenclature, the sets X, Y in your original question are not sequences, so you still haven't given us a definition of bounded that applies to X and Y.

..

..If X and Y are sets though, what is the definition for them to be bounded? It isn't the same as if they were sequences?
I am incredibly confused...

:
A set X is bounded if $\exists M \in \mathbb{R}$ such that $|x| < M\, \forall x \in X$

Isn't this the same thing as what I wrote?

Yes. What I was trying to get *you* to understand is that ghostwalker and I aren't asking for definitions because we don't know the correct one, but because the first thing the OP needs to do is to either state the definition correctly, or at least realise that they don't know the definition (kind of like http://en.wikipedia.org/wiki/Socratic_questioning).

As for the rest of what you wrote - it's pretty close to giving the OP a full solution, which is against the forum rules.