I would like to receive only the hint, how to prove the statement on the heading. I understand that we have to prove that all Cauchy sequences converges in the space $\Bbb R$, e.g. $$\lim_{p,q\rightarrow+\infty}||x_p-x_q||=0\Longrightarrow \exists\lim_{n\rightarrow+\infty}x_n\in \Bbb R.$$
2026-04-22 23:32:41.1776900761
A metric space $(\Bbb R,d)$ with $d(x,y)=||x-y||$ is complete!
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Hint: Every sequence in $R$ contains a monotonic subsequence.