I'm trying to prove that each compact operator is bounded, from the definition:
On the internet I find solutions for this problem, but they always start from a different definition. I would be glad if someone could help me, in finding a proof for this property from this specific definition. Any hints are welcome!
Hint: If operator $A$ is not bounded, there is a bounded sequence $u_n$ such that $\|A u_n\| \to \infty$. How can this contain a convergent subsequence?