I'm trying to prove Corollary 1.4.9 in K. Davidson's book (Exercise 1.5):

If A is a separable C* algebra, then there is an increasing sequence $E_i, i=1,...,\infty$ of positive norm-one elements which form an approximate identity for A.

The hint suggests to choose $E_n$ successively so that $||E_n A_k - A_k|| < \frac{1}{n}$ for all $k$ from $1$ to $n$.

It is clear from how approximate identities are constructed in this book that this can be done with $||E_i||<1$. Why can the $E_i$ be chosen to be increasing and norm = 1?