Inequality Concerning Positive Elements in a C$^{*}$-algebra of Norm at Most One

Question

I am studying the GNS construction and in part of a proof in one of the theorems along the way (theorem 3.3.3. in Murphy's book), the following statement is made:

Now suppose that $a$ is positive and $\|a\|\leq 1$. Then $u_{\lambda}-a$ is hermitian and $\|u_{\lambda}-a\|\leq 1$.

Here, $(u_{\lambda})_{\lambda\in\Lambda}$ is some approximate unit of $A$. The fact that $u_{\lambda}-a$ is self-adjoint is obvious, since both $u_{\lambda}$ and $a$ are positive, but I am unsure how to deduce that $\|u_{\lambda}-a\|\leq 1$. Is it true that in general for two positive elements, each of norm at most one, their difference is of norm at most one?

Thank you.

Answer 1

Since I cannot find the answer (which has given here already a few times) I will just post it: For any two positive elements $a,b$ of norm less or equal to one, we get: $$ -1 \leq -b \leq a - b \leq a \leq 1 $$