Let $A$ be a unital $C^*$-algebra and $A_+$ be the set of positive elements of $A$. Suppose that $(a_n)$ is a sequence of $A_+$. I am trying to prove that
$\lim_{n\to\infty}||a_n||$ exists if and only if $\lim_{n\to\infty}a_n$ exists
I managed to prove the left way of the statement using this norm property: $|\|x\|-\|y\|| \leq \|x-y\|$, but i am stuck while proving the right way. Here is what i have done.
if $\lim_{n\to\infty}||a_n||$ exists, then there is (say) $L\in \mathbb{R}_{\geq0}$ such that $\lim_{n\to\infty}||a_n||=L$. My claim is that $(a_n)$ would converge to $(L)I$ where $I$ is the identity of $A$. So i start with
$||a_n-(L)I||=||a_n-\|a_n\|I+\|a_n\|I-(L)I||\leq||a_n-\|a_n\|I\|+\|\|a_n\|I-(L)I||$
Since $\lim_{n\to\infty}||a_n||=L$, $\|\|a_n\|I-(L)I||=|\|a_n\|-(L)|$ approaches $0$, but i don't know what to do with $||a_n-\|a_n\|I\|$.
is there a clue to prove what i claim or my claim is not true in general?
Apologize if my english is quite a mess and thank you for your help.