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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 $(I+K)x=y$.

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Hint 1: Reduce problem to the case when all $x_n$ are orthogonal to $\mathrm{Ker}(I+K)$.

Hint 2: Prove ad absurdum that ${x_n}$ is bounded. To start this subproof consider vectors $\Vert x_n\Vert^{-1} x_n$.

Hint 3: Find convergent subsequence in $(I+K)x_{n_k}$. Why it does exist?

Hint 4: Show that $\{x_{n_k}\}$ also have a limit (say $x$) and then show $y=(I+K) x\in \mathrm{Im}(I+K)$

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