Show that every subspace of $\mathbb{R}^n$ is closed Show that every subspace of $\mathbb{R^n}$ is closed.
I'm not sure how to do this or even what closed means. I don't even have a starting point. Any hints or solutions are greatly appreciated.

Let $S$ be a linear subspace of $\mathbb{R}^n$. Consider a sequence $\{x_n\}_{n\in\mathbb{N}}$ of points in $S$ which converges to a point $y\in\mathbb{R}^n$. Show that in fact $y$ must lie in $S$. (Here I am using a characterization of "closed" which is equivalent to that described in the other answers).
You could use the continuity of any projection map to the subspace, if that's available to you or easy to prove. You could also show it directly by just doing the analysis. (Hint: the finite-dimensionality should matter)
