Proof of The Earlier and the New Versions of the Definitions of Metric Spaces, Convergence, Continuity and Compactness Agree

Each new version treated here has the text of its earlier version, except that references to definitions treated in this item, to Sequences and to Norms and Real Normed Spaces point to their new versions, and that Metric Space links the words "real numbers" to The Real Numbers as a Type; every other reference, the settings included, is the same. The new sequences agree with the earlier ones by The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (i), and the new real normed spaces with the earlier ones by The Earlier and the New Versions of the Definitions of Euclidean Space, Real Matrices and Normed Spaces Agree (iii). Where a new version still cites an item stated over the earlier notion of metric space, normed space or sequence, that item applies to the same objects with the same meaning, by (i), The Earlier and the New Versions of the Definitions of Euclidean Space, Real Matrices and Normed Spaces Agree (iii) and The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (i). So each condition is the same condition once the notions it cites are known to agree.

(i): Metric Space differs from Metric Space only in the link of "real numbers" to The Real Numbers as a Type. An element of $\mathrm{Real}$ in that sense is given by its set, an element of $\mathbb{R}$ (The Real Numbers as a Type (i)), and corresponds to the element of the earlier $\mathrm{Real}$ with the same set by The Earlier and the New Versions of the Definitions of the Natural Numbers, the Complex Numbers, the Conjugate and the Modulus Agree, and the Two Versions of Real Correspond (v); so in both versions $d$ is a map into $\mathbb{R}$, and the conditions on $d$ are the same.

(ii): by (i) both versions speak of the same metric spaces, and a metric space in the new sense is one in the earlier sense, to which Dense Subset of a Metric Space (i), stated over the earlier notion, applies; both versions ask for a countable subset that is dense in this sense. By The Earlier and the New Versions of the Definitions of Euclidean Space, Real Matrices and Normed Spaces Agree (iii) both speak of the same real normed spaces $V$, and a real normed space in the new sense is one in the earlier sense, so The Norm Metric of a Real Normed Space (i), stated over the earlier notion, gives the same norm metric $d_{V}$ in both. Both ask that $(V,d_{V})$ be separable; by the first part, it is separable in the earlier sense if and only if it is in the new.

(iii): by (i) and The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (i) both versions speak of the same metric spaces and sequences, and the condition that for every real $\varepsilon>0$ there be $N\in\mathbb{N}$ with $d(x_{m},x)<\varepsilon$ for all $m\ge N$ is the same; so convergence to $x$ agrees, and with it being convergent.

(iv): by (iii) both versions speak of the same convergent sequences and define the limit as the unique point to which the sequence converges, the same point by (iii).

(v): by (i) the metric spaces are the same, and a metric space in the new sense is one in the earlier sense, to which Open Ball in a Metric Space (i), stated over the earlier notion, applies; both versions use these open balls, so the condition on $U$ is the same.

(vi): both versions use the same item for Cauchy sequences, which is stated over the earlier notions of metric space and sequence; these agree with the new ones by (i) and The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (i), so both versions have the same Cauchy sequences, and convergence agrees by (iii).

(vii): by (i) the metric spaces are the same, and the $\varepsilon$-$\delta$ condition is the same.

(viii): Compact Topological Space and Compact Subset is Compact Topological Space and Compact Subset word for word, with the same references.