Prove set is dense This is a pretty basic and general question.
I have to prove (if true) that the sum of two dense sets is dense as well.

Let $A, B$ be nonempty dense sets in $\mathbb R$. Then $A+B=\{a+b\mid a\in A, b\in B\}$ is also dense.

Can anyone give me a pointer as to how one may prove this (just the method)? is it algebraic or with $\sup /\inf/\text{sequences}$ etc.
Thanks!

Hint: 


*

*If $A$ is dense in $\mathbb{R}$, then $A+2013 = \{a+2013 \mid a \in A\}$ is also dense in $\mathbb{R}$.

*Union of dense sets is dense, in particular, $A+B = \bigcup_{b \in B}A+b$.


Good luck!
