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Find a Cauchy sequence in a C*-algebra

I have an Example of cauchy sequence in a Hilbert space. Now I am working in a C*-algebra $A$. Suppose that $\{x_n\}$ is a sequence in $A$ such that $\parallel x_n \parallel \leq \frac{1}{n}$, $\parallel x_n + x_m \parallel \leq 1$ and $\parallel x_nx_m \parallel \leq \frac{1}{n}$.
I have to find a Cauchy sequence.
I tried to use the triangle inequality but there is no guarantee that it is true.
How can I show the existence of a cauchy sequence? I know that $A$ is a Banach algebra.
Thanks

A:

If $\{x_n\}$ is a Cauchy sequence in $A$, then there exists $a\in A$ such that $\parallel a-x_n\parallel\to0$ as $n\to\infty$. Then
$$\parallel a-x_n\parallel\leq\parallel a\parallel+\parallel x_n\parallel\leq 2\parallel a\parallel\to0$$
Thus the sequence $\{x_n-a\}$ is also Cauchy.

On the infrared spectrum of the superconducting pnictide phase BaFe2As2.
The infrared (IR) spectrum of BaFe(2)As(2) in the crystalline phase is reported. In addition to the previously reported renormalizations of the phonon spectrum, the new experimental results reveal the presence of negative Grüneisen parameters for selected phonon modes which are not affected by the static electric field. These modes are in agreement with the notion that the Fe atoms are fluctuating between
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