## Why is $\P_d$ closed in $C([a,b])$?

Recall that $\P_d$ is the set of all real polynomials of degree at most $d$, and we consider the space $C([a,b])$ equipped with the $\sup$ norm. The claim is that $\P_d$ is a closed subset of $C([a,b])$. This is actually quite tricky to show—at least, I don’t know of a simple argument. Here is what…