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Let be $\{ s_n \}$ a Cauchy sequence and $\{s_{n_k}\}$ is a subsequence that belongs to $\{ s_n \}$ such that $\lim s_{n_k} = a$ then $\{ s_n \}$ converge to $a$.



Given $\varepsilon > 0$, there exists $ n_0 \in \mathbb{N}$ such that $m,n > n_0 \Longrightarrow |x_n-x_m| < \frac{\varepsilon}{2} $. There exists $n_k > n_0 $ too, such that $|s_{n_k} - a | < \frac{\varepsilon}{2}$. Therefore, $n > n_0 \Longrightarrow |s_n-s_{n_k}| + |s_{n_k} - a| < \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon$. This show us that $\{ s_n \}$ converges to $a$