I am reading Rordam's paper "On the Structure of Simple C$^{*}$-Algebras Tensored with a UHF-Algebra." I came across the following passage:
I am having trouble understanding the $(\impliedby)$ direction in (2.3) and also the statement (2.4).
The $(\implies)$ direction in (2.3) is easy: if $a<\alpha$, then, given $e\in GL(A)$, we have $$ \|x-e\|\leq\|x-x_{a}\|+\|x_{a}-e\|\leq a+\|x_{a}-e\|, $$ which implies that $$ \|x_{a}-e\|\geq \|x-e\|-a\geq\alpha-a>0. $$ Thus, $x_{a}\notin\overline{GL(A)}$.
I would really appreciate some help with the other direction and (2.4). It is probably quite simple and I must have forgotten something, but I have been stumped for a while.
EDIT:
As André S. points out in the comments, the reverse implication in (2.3) follows from (2.4), so
all that remains is to prove (2.4).
Thank you.