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!
2026-04-01 19:45:04.1775072704
Why is a linear transformation of a cauchy sequence in a normed space also cauchy?
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Given $n,m\in \mathbb N; \exists p\in \mathbb N $ such that $||a_n-a_m||<\epsilon \forall m,n\geq p$
Now $||T(a_n)-T(a_m)||=||T(a_n-a_m)||<\epsilon$