Image of orthonormal basis under compact operator

Question

Let T be a compact operator on a Hilbert space H. Then we need to prove that the sequence formed on taking the image of an orthonormal basis goes to 0. I read a proof which used the fact that every compact operator can be written as the limit of a sequence of finite rank operators, which I think is incorrect.

Answer 1

You can look here. The fact that you're taking a basis does not change anything respect to the answer linked.