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A constant sequence has the same value $a$ at every index. It converges to $a$ by convergence of a sequence in a topological space.

Constant sequences give an injection $\mathbb{R}\to {}^{\omega }\mathbb{R}$. For the opposite bound fix an enumeration $(q_m{)}_{m\in \omega }$ of the rational numbers and encode a sequence $(x_n)$ byDistinct sequences differ at some coordinate $n$; density of the rational numbers then makes their encoded sets differ at a pair $(n,m)$. Thus $E$ is injective. The Cartesian product $\omega \times \omega$ is countable; for example the Cantor pairing functionis a bijection with $\omega$. Consequently $\mathcal{P}(\omega \times \omega )$ is equinumerous with $\mathcal{P}(\omega )$, already equinumerous with $\mathbb{R}$. Apply Cantor-Schröder-Bernstein theorem to obtain

By convergence of a sequence in a topological space, $x_n\to a$ means that every open neighbourhood of $a$ contains all sufficiently late terms. In the cofinite topology, this is equivalent to the condition that, for every $b\ne a$, the value $b$ occurs only finitely often. Necessity uses the open neighbourhood $X\setminus \{b\}$; sufficiency chooses an index beyond every last occurrence of the finitely many points excluded by a given open neighbourhood.

Consequently a sequence with distinct terms converges to every point of $X$. For the original choice $x_n=n$ on $X={\mathbb{N}}_{>0}$, take the two open neighbourhoodsAll terms with $n\ge 9$ lie in $U_1$ and all terms with $n\ge 8$ lie in $U_2$. More generally, each finite excluded set has a last visit, so $x_n\to 1$ and $x_n\to 2$, and in fact $x_n\to a$ for every $a\in X$. No numerical distance is tending to zero here; the open sets specify the meaning of convergence.

Distinct terms are sufficient, not necessary. The sequence $1,1,2,2,3,3,\dots$ also converges to every point, since each value occurs only twice. By contrast, the constant sequence $x_n=1$ converges only to $1$: an open neighbourhood of any $a\ne 1$ can exclude $1$. The alternating sequence $1,2,1,2,\dots$ converges to no point, since some value different from any proposed limit occurs infinitely often.