(Quick note: I see there are a lot of Cauchy sequence questions but I did not see this question specifically)
Suppose that the sequence $v_n, n=1,2,3,... $ of elements from an inner product space $V$ converges to $v\in V$. Prove that $v_n$ is a Cauchy sequence.
I use the definition of the Cauchy sequence:
$$|v_n - v_m| = |(v_n-v) + (v-v_m)|$$
Using the triangle inequality,
$$|(v_n-v) + (v-v_m)| \leq |v_n-v| + |v_m-v|$$
By the given, it is clear that $|v_n-v|<\epsilon_0$ given any $n>N$. I'm not really sure what to do with the $|v_m-v|$ part, though. Is it enough to say that $|v_m-v|>0$, and then to call this scalar $\epsilon _1$. Then we can say that $$|v_n-v_m|< \epsilon _0 + \epsilon _1$$ And, since both of these $\epsilon$ values are finite and greater than zero, then their sum is as well, so we set $\epsilon = \epsilon _0+ \epsilon_1$ and see that the definition of the Cauchy sequence is satisfied?