Let $ \mathcal{H} $ be a Hilbert space and $ M $ be a closed subspace, prove that $ M + x_0\mathbb{R} $ is a closed subspace.
It was easy enough to prove that $ M + x_0\mathbb{R} $ is a vector subspace, but I got into trouble trying to prove it's closed.
I tried defining the sequence $ x_n + a_n x_0 $ (where $ x_n \in M $ and $ a_n \in \mathbb{R} $) assuming it converges to $ v $, in order to show that $ v\in M + x_0\mathbb{R} $. My approach is to show that both $ x_n $ and $ a_n $ have to be Cauchy sequences in their respective subspace. Doing so, it's easy to imply that $ v = \lim x_n + x_0 \lim a_n$. However, I wasn't able to do so (and to be completely honest, I'm not even convinced that has to hold, but I could not come up with any other approach).