we studied that in any metric spaces cauchy sequences are bounded. and we know that cauchy sequence converges iff it has a convergent subsequence. when we prove that $$R$$ is a banach space we had used these two theorem and Bolzano- Weierstrass theroem. that says " Every bounded sequence has a convergent subsequence." My question is: as this theroems is valid for any metric space then we can use them to prove that in any metric space cauchy sequences are convergent as we have did in $$ R$$ in fact we know this is not true for example $$ (1+ \frac{1}{n})^n \subset Q $$ but $$ (1+ \frac{1}{n})^n \to e\notin Q $$ so it isn't convergent sequence in Q then my prove is false but I want to know what is wrong with this proof?
2026-04-01 16:27:50.1775060870
cauchy sequences and convergence of this sequences
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