The square roots $\sqrt{k}$ diverge to $+\mathrm{\infty }$

Given: For every natural $k$ the canonical natural $k\cdot 1_{\mathbb{R}}$, written $k$, and the real $\sqrt{k}$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences, Canonical naturals are positive and strictly increasing).

Divergence to $+\mathrm{\infty }$: $z_k\to +\mathrm{\infty }$ means that for every real $M$ there is $K$ with $z_k>M$ for all $k\ge K$ (Divergence to $+\mathrm{\infty }$ and to $-\mathrm{\infty }$).

Archimedean property: for every real $x$ there is a natural number $n\ge 1$ with $x<n\cdot 1_{\mathbb{R}}$ (Every complete ordered field is Archimedean).

Canonical naturals: $0\cdot 1_{\mathbb{R}}=0$ and $n\cdot 1_{\mathbb{R}}>0$ for $n\ge 1$; the map $n\mapsto n\cdot 1_{\mathbb{R}}$ is strictly increasing, hence preserves non-strict order, on $\{1,2,3,\dots \}$; so the real $k=k\cdot 1_{\mathbb{R}}$ is nonnegative for every natural $k$ (The unique embedding of ℚ into an ordered field, Canonical naturals are positive and strictly increasing).

Square roots: every real $a\ge 0$ has a unique $s\ge 0$ with $s^2=a$, written $s=\sqrt{a}$ (Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$).

Monotonicity of squaring on nonnegatives: for $a,b\ge 0$, $a<b⟺a^2<b^2$ (Squaring is monotone on the nonnegatives).

Trichotomy and order arithmetic: for all reals $u,v$ exactly one of $u<v$, $u=v$, $v<u$ holds, and $u\ge 0>v$ implies $u>v$ (Ordered field, Order on the reals).

A sequence converging to a real is bounded, so a sequence that diverges to $+\mathrm{\infty }$ — being unbounded, since it eventually exceeds every real bound — is not convergent (Every convergent sequence is bounded, Divergence to $+\mathrm{\infty }$ and to $-\mathrm{\infty }$, Sequences of reals: bounded, eventually, frequently, tails, subsequences).