Why is a linear transformation of a cauchy sequence in a normed space also cauchy? Suppose we have a cauchy sequence $\{a_n\}$ in a normed vector space $V$. Given a linear transformation $T:V \rightarrow V$, is the sequence $\{T(a_n)\}$ also cauchy? Or is it true only for finite dimensional normed spaces? I'd be much obliged if someone could give a proof for this, preferably an elementary one.
Thanks!
Given $n,m\in \mathbb N; \exists p\in \mathbb N $  such that $||a_n-a_m||&lt;\epsilon \forall m,n\geq p$

Now $||T(a_n)-T(a_m)||=||T(a_n-a_m)||&lt;\epsilon$
Given $n,m\in \mathbb N; \exists p\in \mathbb N $  such that $||a_n-a_m||&lt;\epsilon \forall m,n\geq p$

Now $||T(a_n)-T(a_m)||=||T(a_n-a_m)||&lt;\epsilon$