Show that if $(x_{n})_{n}$ is a Cauchy sequence in X and $\lambda \in \mathbb{R}$, then the sequence $(\lambda x_{n})_{n}$, is also Cauchy in X.
We know that for $(x_{n})_{n}$, we have $\forall \epsilon >0:\exists N\in \mathbb{N} : n,m\ge N\implies ||x_{n}-x_{m}||\le \epsilon$
We can also assume that $||\lambda (x_{n}-x_{m})||\le \epsilon$
So to prove this, we can say that:
$||\lambda (x_{n}-x_{m})||\le |\lambda |\cdot||x_{n}-x_{m}|| \le |\lambda|\epsilon$
But I can't help but feel dubious about having the $\lambda$ at the end. Any tips?