Need a help in understanding a solution of a problem on Compact operators.

Question

The question and its solution is given in the following pictures:

But I could not understand the statement in the third line from below that say:"But since $A^{n-2} \in L(H)$ and $A^n$ is compact, $A^{2n-2}$ is also compact ", My problem is with the power of the operator used,

1- why we are sure that $A^{n-2} \in L(H)$?

2- why $A^{n-2} \in L(H)$ and $A^n$ is compact leads that $A^{2n-2}$ is also compact?

Could anyone explain this for me ?

Thanks!

Answer 1

1) Any positive integer power of a bounded linear operator is a bounded linear operator.

2) The product of a bounded linear operator and a compact linear operator is compact.