Approximating the limit of a Cauchy sequence in a Banach space Let $E$ be a Banach space and consider a sequence $(x_n)_n$ in $E$ satisfying the following condition: $$||x_n-x_{n-1}||\leq 3^{-n}\mbox{ for all }n\in\mathbb{N}.$$
Clearly $(x_n)_n$ is a Cauchy sequence and therefore converges to an element $x\in E$. 
Question: How to prove that $||x-x_n||\leq \frac{1}{2}3^{-n}$ for all $n\in\mathbb{N}$?

We can prove it using triangle equation:
$$||x-x_n|| \leq \sum_{k=n}^{\infty}||x_{k+1}-x_k|| \leq \sum_{k=n}^{\infty}3^{-k-1}=\frac{3^{-n-1}}{1-\frac{1}{3}} = \frac{3^{-n}}{2}$$
