The sum of two divergent sequences can converge: $k+(-k)$

Given: The sequences $x_k:=k$ and $y_k:=-k$, where the natural $k$ denotes its canonical image $k\cdot 1_{\mathbb{R}}$ in $\mathbb{R}$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences, Canonical naturals are positive and strictly increasing).

Divergence to signed infinity: $z_k\to +\mathrm{\infty }$ when for every real $M$ there is $K$ with $z_k>M$ for all $k\ge K$; and $z_k\to -\mathrm{\infty }$ when 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: for every $n\ge 1$ the element $n\cdot 1_{\mathbb{R}}$ is positive, and $n\mapsto n\cdot 1_{\mathbb{R}}$ is strictly increasing on $\{1,2,3,\dots \}$, hence preserves non-strict order there (Canonical naturals are positive and strictly increasing).

Order arithmetic in an ordered field: $a<b$ means $b-a$ is positive, so $a<b$ if and only if $-b<-a$; and trichotomy holds for the strict order (Ordered field, Order on the reals).

Field arithmetic: every element $u$ has an additive inverse $-u$ with $u+(-u)=0$, and $u-u=0$ (Field).

Absolute value: $\mathrm{\mid }u\mathrm{\mid }=u$ whenever $u\ge 0$, so $\mathrm{\mid }0\mathrm{\mid }=0$ (Absolute value in an ordered field).

Convergence: a sequence $(z_k)$ converges to $z$ when for every rational $\epsilon >0$ there is $K$ with $\mathrm{\mid }{z}_k-z\mathrm{\mid }<\epsilon$ for all $k\ge K$ (Limits and Cauchy sequences of reals).

Every convergent sequence is bounded: if $(z_k)$ converges to $z$ then there is $M$ with $\mathrm{\mid }{z}_k\mathrm{\mid }\le M$ for every $k$ (Every convergent sequence is bounded).

Boundedness: $(z_k)$ is bounded when there is a real $M$ with $\mathrm{\mid }{z}_k\mathrm{\mid }\le M$ for every $k\in \mathbb{N}$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences).

A sequence diverging to $+\mathrm{\infty }$ or to $-\mathrm{\infty }$ is unbounded, and an unbounded sequence is not convergent — the latter being the contrapositive of [F8] (Divergence to $+\mathrm{\infty }$ and to $-\mathrm{\infty }$, Every convergent sequence is bounded, Sequences of reals: bounded, eventually, frequently, tails, subsequences).

Sum rule of the algebra of limits: if $(u_k)$ converges to $u$ and $(v_k)$ converges to $v$, then $u_k+v_k$ converges to $u+v$ (Algebra of limits: sums, scalar multiples, products and quotients).