Proving a linear operator is compact: understanding the statement "norm limit of a sequence of finite rank operators".

Question

I am having serious trouble understanding the proof that an operator is compact.

This is the original question I asked and the proof is included very helpfully in the answer.

Show if $\lim_{n \to \infty} \lambda_n=0$ then $Tu=\sum^\infty_{n=1} \lambda_n \langle u,e_n \rangle e_n$ defines a compact operator.

When showing the operator $T$ is compact the main criteria mentioned is "show $T$ is the norm limit of a sequence $T_n$ of finite rank operators".

What does this statement mean?

Do we have a finite rank operator?

Apologies if I am not asking the right question because I am not sure where to begin.

In the proof, why do we have to show $$||T-T_k || \to 0$$

Is it because this shows that $T$ is the limit of a sequence of finie rank operators? So $T_k$ are the finite rank operators and $T$ is the limit?