[Math] Every Cauchy sequence in a normed vector space is bounded

Question

But what about Cauchy sequences in non-normed vector spaces? How can we even measure a vector in a sequence if there is no norm?

Answer 1

a cauchy sequence inside any metric space is bounded.

Notice that there exists a natural number $N$ such that $d(X_N,x_m)<1$ for all $m>N$.

Let $D=\max\limits_{i<N}d(x_N,x_i)$. Then the ball of radius $D+1$ centered at $X_N$ contains all of the sequence.