The alternating sequence $(-1{)}^k\mathrm{/}(k+1)$ converges to $0$

Given: The unique sequence $(s_k)$ of reals with $s_0=1$ and $s_{\sigma (k)}=-s_k$ (The recursion theorem), the sequence $y_k:=s_k(k+1{)}^{-1}$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences), and the sequence $u_k:=(k+1{)}^{-1}$.

The recursion: $(s_k)$ is the unique sequence with $s_0=1$ and $s_{\sigma (k)}=-s_k$ for every $k$ (FALSE: every bounded sequence converges).

Induction principle: if $P(0)$ holds and $P(k)$ implies $P(\sigma (k))$ for every $k$, then $P(k)$ holds for every natural $k$ (The principle of mathematical induction).

Absolute value is even: $\mathrm{\mid }-u\mathrm{\mid }=\mathrm{\mid }u\mathrm{\mid }$ (Basic properties of the absolute value).

Absolute value: $\mathrm{\mid }u\mathrm{\mid }=u$ if $u\ge 0$ and $\mathrm{\mid }u\mathrm{\mid }=-u$ if $u<0$ (Absolute value in an ordered field).

$(u_k)=((k+1{)}^{-1})$ converges to $0$ and every term is positive (The sequence $1\mathrm{/}(k+1)$ converges to $0$).

Null times bounded: if $(z_k)$ converges to $0$ and $(w_k)$ is bounded, then $(z_k{w}_k)$ converges to $0$ (A null sequence times a bounded sequence is null).

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).

$0<1$ in $\mathbb{R}$ (The multiplicative identity is positive).