# Translation of a dense set in a normed vector space

I would like to prove that if $X$ is, let's say, a normed vector space, and $A$ is a dense subset of $X$, and $x_0 \in X$, that $\{x_0\} + A$ is also dense in $X$.

I know that translation by a point is a homeomorphism, but not translation by a set. I have also tried using sequences, and using closure arguments. Is there any easy way to do this problem?

A set is dense if it meets every non-empty open set. $V$ meets $x_0+A$ iff $V-x_0$ meets $A$.

• Of course. Thanks! – Johnny Apple Sep 23 '15 at 2:52
• @Johnny: You're welcome! – Brian M. Scott Sep 23 '15 at 3:00

An elementary proof, using sequences, is the following:

Let $$x\in X$$.

$$A$$ is dense and therefore there exists a sequence $$\{x_n\}$$ in $$A$$ such that $$x_n\to x-x_0$$. So the sequence $$y_n=x_0 + x_n$$ is in $$\left \{ x_0 \right \}+A$$ and $$y_n\to x_0+x-x_0=x$$. So $$x\in\overline{ \left \{ x_0 \right \}+A}$$ and therefore $$\left \{ x_0 \right \}+A$$ is dense.