Convergence and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space

Let $\mathbb{R}$ denote the real numbers, with the addition, multiplication, identities, additive inverses and order of their ordered field structure; for $s,t\in\mathbb{R}$ write $s-t$ for $s+(-t)$, write $s<t$ to mean that $s\le t$ and $s\ne t$, let $|s|$ be the absolute value of $s$, and let $d_{\mathbb{R}}$ be given by $d_{\mathbb{R}}(s,t)=|s-t|$, which is a metric on $\mathbb{R}$ by The Absolute Value Metric on the Real Line, so that $(\mathbb{R},d_{\mathbb{R}})$ is a metric space. Let $\mathbb{N}$ be the set of natural numbers.

The notation $(a_n)_{n=1}^{\infty}$ used in Limit of a Sequence of Real Numbers and in Cauchy Sequence of Real Numbers is not given a formal set-theoretic meaning in those items. We therefore stipulate the following reading of them, which is in force throughout this lemma and its proof: $(a_n)_{n=1}^{\infty}$ denotes a sequence $(a_m)_{m\in\mathbb{N}}$ in $\mathbb{R}$ indexed by $\mathbb{N}$; the indices $n$ quantified over in those items are integers, and an inequality $n\ge N$ between such an integer $n$ and an element $N$ of $\mathbb{N}$ is read in $\mathbb{R}$ through the canonical map $\iota:\mathbb{N}\to\mathbb{R}$, that is, as $\iota(N)\le n$; and when an integer $n$ has the form $n=\iota(m)$ with $m\in\mathbb{N}$, which determines $m$ uniquely by claim 7 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, the term $a_n$ denotes $a_m$. This reading is a stipulation about notation, not a fact imported from elsewhere.

Let $(a_m)_{m\in\mathbb{N}}$ be a sequence in $\mathbb{R}$ and let $L\in\mathbb{R}$. Then the following hold.

1. (Convergence) The sequence $(a_m)_{m\in\mathbb{N}}$ converges to $L$ as a sequence of real numbers if and only if it converges to $L$ in $(\mathbb{R},d_{\mathbb{R}})$.

2. (Cauchy condition) The sequence $(a_m)_{m\in\mathbb{N}}$ is a Cauchy sequence of real numbers if and only if it is a Cauchy sequence in $(\mathbb{R},d_{\mathbb{R}})$.