I have to prove that if $A$ is a C*-Algebra then the algebra $A_1$ obtained adjoining the identity is a C*Algebra too (with the usual algebraic operation defined).
I have any problem in all the points of the proof but in one: to prove that the new space is Banach.
The norm in the new space is defined to be:
$\|a+\alpha\|= \sup\{\|ax + \alpha x\|: x\in A, \|x\|\leq 1\}$
I have to prove that every cauchy sequence $(a_n+\alpha_n)_n$ is convergent. Intuitively it exist a limit for $a_n$ that is $a$, and a limit for $\alpha_n$, that is $\alpha$. Then I have to show that $\lim \sup_{n\rightarrow\infty}\{\|a_nx + \alpha_n x -ax - \alpha x\|: x\in A, \|x\|\leq 1\}=0$. But the formal proof escapes me. Can you give me a little help? Thanks.