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?
2026-03-29 22:26:49.1774823209
Every Cauchy sequence in a normed vector space is bounded
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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.