Cauchy sequences are bounded in every metric space

Question

A few days laid out an example, and asked for help, and @ shadow10 replied, his answer the question of can I find the

https://math.stackexchange.com/questions/879662/every-Cauchy-sequence-is-bounded

but please someone help me in relation to this question: as would seem replies @ shadow10 in a space of arbitrary metric.

I know only that we should place absolute value $d (x, y)$.

Please help. Previously thank you

Answer 1

Let $(X,d)$ a metric space and $(x_n)_n$ a Cauchy sequence, and $a\in X$. $\epsilon=1$. There is a $N$ such that $\forall m,n\ge N$ $$d(x_m,x_n)<1$$ by triangular inequality we have $d(x_n,a)\leq d(x_n,x_N)+d(x_N,a)$, then for $n\geq N$ we obtain $d(x_n,a)\leq 1+d(x_N,a)=C$.

Then If we take $M=\max\{d(x_0,a),\dots,d(x_{N-1},a),C\}$. Then $d(x_n,a)\le M\;\forall\,n\in \mathbb{N}$.