I want to prove that the direct sum of two (complex) Hilbert spaces is a Hilbert space. I've shown that we have an inner product and also shown norm however I have trouble to show converges. We define our Hilbert spaces as follows, let $H_1,H_2$ be Hilbert spaces, then the direct sum $H_1\oplus H_2$ is defined below $$\langle (x,y)|(x',y')\rangle:=\langle x|x’ \rangle_{H_1}+\langle y|y’ \rangle_{H_2} ,$$ where $(x,y),(x’,y’)\in H_1\times H_2 $.
Yet I have readed a proof for a similar question however I didn't get so much from it. Here is the link. Countable family of Hilbert spaces is complete