If $X$ is a non empty closed subset of $\mathbb{R}$ and $Y = [1,2]$ then $X+Y$ is closed.

Question

If $X$ is a non empty closed subset of $\mathbb{R}$ and $Y = [1,2]$ then $X+Y$ is closed?

How to show this?

I know that sum of two closed set may not be closed. But how to prove this? Can anyone give me a hint?

Answer 1

Suppose $x_n \in X,y_n \in [1,2]$ and $x_n +y_n \to z$. Since $[1,2]$ is compact there is a subsequence $y_{n_{k}}$ which converges to some point $y$ in $[1,2]$. But $x_{n_{k}}+y_{n_{k}} \to z$ so $x_{n_{k}} \to z-y$. Since $X$ is closed, $z-y \in X$. It follows that $z= \lim (x_{n_{k}}+y_{n_{k}}) =(z-y)+y \in X+Y$.

Answer 2

And an answer to the second part of your question. Let $X=\{2,3,4,\ldots\}$, $Y=\{y:y=-n+\frac1n, n=2,3,4,\ldots\}$. They are closed, but in their sum there is a set $\{\frac1n:n=2,3,4, \ldots\}$ but not $0$.