Suppose $K:H\to H$ is a compact linear operator on a Hilbert space $H$.
How do I show that the range of $I+K$ is closed in $H$? I believe this is equivalent to showing that $\{x_n\}\subset H$ and $(I+K)x_n\to y\in H \implies \exists x\in H$ such that $(I+K)x=y$.