# Prob. 25(b), Chap. 4 in Baby Rudin: The set $C_1 + C_2$ need not be closed in $\mathbb{R}$ even for closed sets $C_1$ and $C_2$

Here is Prob. 25, Chap. 4 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition:

If $A \subset \mathbb{R}^k$ and $B \subset \mathbb{R}^k$, define $A + B$ to be the set of all sums $x+y$ with $x \in A$, $y \in B$.

(a) If $K$ is compact and $C$ is closed in $\mathbb{R}^k$, prove that $K+C$ is closed. [This I think I've managed to prove.]

Hint: . . .

(b) Let $\alpha$ be an irrational real number. Let $C_1$ be the set of all integers, let $C_2$ be the set of all $n \alpha$ with $n \in C_1$. Show that $C_1$ and $C_2$ are closed subsets of $\mathbb{R}^1$ whose sum $C_1 + C_2$ is not closed, by by showing that $C_1 + C_2$ is a countable dense subset of $\mathbb{R}^1$.

My effort:

Let $p$ be a real number that is not an integer, and let $\delta$ be any real number such that $$0 < \delta < \min \left\{ \ p - \lfloor p \rfloor, \ \lceil p \rceil - p \ \right\}.$$ Then the $\delta$-neighborhood of $p$ in $\mathbb{R}^1$ does not intersect the set of integers at all, showing that every point $p$ of $\mathbb{R}^1 - C_1$ is an interior point and hence that $C_1$ is closed in $\mathbb{R}^1$. Am I right?