Proof of Convergence, Limits and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space, which is Complete

By The Distance on the Real Numbers §distance, $d_{\mathbb{R}}(s,t)=|s-t|$ for all $s,t\in\mathbb{R}$; in particular $d_{\mathbb{R}}(a_{n},L)=|a_{n}-L|$ and $d_{\mathbb{R}}(a_{m},a_{n})=|a_{m}-a_{n}|$ for all $m,n\in\mathbb{N}$.

Converges. For a real $\varepsilon>0$ and $N\in\mathbb{N}$, the condition "$|a_{n}-L|<\varepsilon$ for every $n\in\mathbb{N}$ with $n\ge N$" required in Convergent Sequences of Real Numbers §converges and the condition "$d_{\mathbb{R}}(a_{n},L)<\varepsilon$ for every $n\in\mathbb{N}$ with $n\ge N$" required in Convergent Sequence in a Metric Space §converges for the metric space $(\mathbb{R},d_{\mathbb{R}})$ are the same condition, by the identity above. Both definitions ask that for every real $\varepsilon>0$ there be such an $N$, so each holds exactly when the other does.

Limit. Let $(a_{n})$ be convergent and let $A$ be its limit, the unique real number to which it converges. By the clause converges, $(a_{n})$ converges to $A$ in $(\mathbb{R},d_{\mathbb{R}})$, so it is convergent in $(\mathbb{R},d_{\mathbb{R}})$ by Convergent Sequence in a Metric Space §convergent. Its limit in $(\mathbb{R},d_{\mathbb{R}})$ is, by The Limit of a Convergent Sequence in a Metric Space §limit, the point to which it converges in $(\mathbb{R},d_{\mathbb{R}})$, unique by Uniqueness of Limits in a Metric Space §unique; hence it is $A$.

Cauchy. For a real $\varepsilon>0$ and $N\in\mathbb{N}$, the condition "$|a_{m}-a_{n}|<\varepsilon$ for all $m,n\in\mathbb{N}$ with $m,n\ge N$" required in Cauchy Sequences of Real Numbers §cauchy and the condition "$d_{\mathbb{R}}(a_{m},a_{n})<\varepsilon$ for all $m,n\in\mathbb{N}$ with $m,n\ge N$" required in Cauchy Sequence in a Metric Space §cauchy are the same condition, and both definitions ask for such an $N$ for every real $\varepsilon>0$; so each holds exactly when the other does.

Complete. Let $(b_{n})$ be a Cauchy sequence in $(\mathbb{R},d_{\mathbb{R}})$. By the clause cauchy, applied to $(b_{n})$, it is a Cauchy sequence in the sense of Cauchy Sequences of Real Numbers §cauchy, so it is convergent by Completeness of the Real Numbers for Sequences: Monotone Convergence, the Bolzano-Weierstrass Theorem and Cauchy Sequences §cauchy: it converges to some $B\in\mathbb{R}$. By the clause converges, applied to $(b_{n})$ and $B$, it converges to $B$ in $(\mathbb{R},d_{\mathbb{R}})$, so it is convergent in $(\mathbb{R},d_{\mathbb{R}})$ by Convergent Sequence in a Metric Space §convergent. Hence $(\mathbb{R},d_{\mathbb{R}})$ is complete by Complete Metric Space §complete.