Let $X, Y$ be subsets of $\mathbb{R}^{n}$ and define $X+Y=\{x+y: x \in X, y \in Y\}$. For each of the following statements give a proof or a counterexample (with justification) as appropriate.

(i) If each of $X, Y$ is bounded and closed, then $X+Y$ is bounded and closed.

(ii) If $X$ is bounded and closed and $Y$ is closed, then $X+Y$ is closed.

(iii) If $X, Y$ are both closed, then $X+Y$ is closed.

(iv) If $X$ is open and $Y$ is closed, then $X+Y$ is open.

[The Bolzano-Weierstrass theorem in $\mathbb{R}^{n}$ may be assumed without proof.]