Hello,

in the book functional analysis by reed, there is a lemma which says:

If $\displaystyle A \in L(H)$ is a bounded linear operator on a Hilbert Space, and$\displaystyle A\ge0$. Then there is a unique $\displaystyle \( B \in L(H) \) B\ge0$ s.t. $\displaystyle B^2 = A.$

I have a question about the proof. The author argues:

"It is sufficient to consider the case where$\displaystyle \|A\| \ge 1$."

Why is it sufficient to proof just this case?

I couldn't find the right argument.

Regards