Prove divergence of an alternating sequence

Prove that the divergence of the following sequence.$$s_n=\frac{(-1)^nn}{2n-1}$$

The following is the sample answer

Note that $\exists N\in\mathbb{N}\, s.t.\,\forall k\geq N$$$|s_{2k}-1/2|<1/2$$$$|s_{2k+1}+1/2|<1/2$$

So$$l\geq 0\implies|s_{2k+1}-l|>1/4$$$$l<0\implies|s_{2k}-l|>1/4$$

$s_n$ diverges.

First I do not know where the above four inequalities come from.Second I do not know why the last two inequalities implies that $s_n$ diverges.For me, what I will try to prove alternating sequence diverge is suppose that $s_n$ will converge to $s$ and prove that $s$ does not exist to show contradiction. Or, I may prove that $\lim \sup(s_n)\neq \lim \inf(s_n)$