$X$ open, $X+Y$ also open

Question

Question: Let:

$$X,Y \subset\mathbb{R}$$

and:

$$X+Y= \{x + y : x\in X, y \in Y\} $$

Show that if $X$ is open, then $X+Y$ is also open.

I'm not sure where to start can someone help me it would be appreciated.

Answer 1

Take $z\in X+Y$. There are $x$ and $y$ such that $z=x+y$. As $X$ is open, there is $r>0$ such that $(x-r,x+r) \subset X$. We then have: $$ (z-r, z+r)= (x+y-r, x+y+r) = (x-r,x+r)+\{y\} \subset X+Y.$$

Answer 2

Let bet $x\in X$, as $X$ is open you know that exists $\epsilon$ such that $(x-\epsilon, x+\epsilon)\subset X$. Then you have $(x+y-\epsilon, x+y+\epsilon)\subset X+Y$ for all $y\in Y$.