Question about sum of an alternating series 
Question if $s$ is sum of the alternating series $\sum (-1)^{n+1}z_n$ and if $s_n$ is the nth partial sum then, $|s-s_{n}|≤z_{n+1}$

I had asked this question already here
Is the sum of an alternating series satisfies the following inequality
but, i think answer that posted is not correct.
How is assumed that, $(z_n)$ is monotonic?
Answer goes like this
$(s - s_{n}) = \sum_{k=1}^{\infty} (-1)^{k+1} z_{k} -\sum_{k=1}^{n} (-1)^{k+1} z_{k}   = z_{n+1} -( ( z_{n+2} - z_{n+3}) + (z_{n+4} - z_{n+5})+( z_{n+6} - z_{n+7}) + ............................      ) $
How he know $(-1)^{n+2}z_{n+1}=z_{n+1}$  is $n+2$ is even? This is not mentioned anywhere. Something wrong with this answer. Please help me in Question.
 A: Subhajit has already proved that there are sequences $\left\{z_n\right\}_{n\in\mathbb{N}}$ of non-negative real numbers such that the alternating series $\sum_\limits{n=1}^{\infty}(-1)^{n+1}z_n$ is convergent and the inequality $|s-s_n|\le z_{n+1}$ is wrong.
For example, $z_n = \begin{cases}2^{-n} & \text{ n is even} \\ 3^{-n} & \text{ n is odd }\end{cases}$.
So if we wish the inequality would always hold without exceptions, it is necessary to add some hypothesis. I am going to add monotony, but if you do not like it, in any case, you have to add some other hypothesis in order to prove the inequality.
Let $\left\{z_n\right\}_{n\in\mathbb{N}}$ a sequence of non-negative real numbers such that $z_n\ge z_{n+1}\;\forall n\in\mathbb{N}\;\;\text{and}\;\lim_\limits{n\to\infty}z_n=0.\\\text{Let }s\text{ be the sum of the alternating series}\;\sum_\limits{n=1}^{\infty}(-1)^{n+1}z_n\;\text{and}\\\text{let } s_n \text{ be the } n^{th} \text{ partial sum.}$
Since $\;z_{2m+1}\ge z_{2m+2}\;\forall m\in\mathbb{N},\;$ it follows that
$s_{2m+2}-s_{2m}=z_{2m+1}-z_{2m+2}\ge0\;\;\forall m\in\mathbb{N},\;$ therefore
$s_{2m+2}\ge s_{2m}\;\;\forall m\in\mathbb{N}$.
Hence, $\;\;s_{2m}\le s_{2m+2}\le \sup_\limits{m\in\mathbb{N}}\left\{s_{2m}\right\}=\lim_\limits{m\to\infty} s_{2m}=s.\;\;(*)$
Since $\;z_{2m}\ge z_{2m+1}\;\forall m\in\mathbb{N},\;$ it follows that
$s_{2m+1}-s_{2m-1}=z_{2m+1}-z_{2m}\le0\;\;\forall m\in\mathbb{N},\;$ therefore
$s_{2m+1}\le s_{2m-1}\;\;\;\forall m\in\mathbb{N}$.
Hence, $\;\;s=\lim_\limits{m\to\infty} s_{2m-1}=\inf_\limits{m\in\mathbb{N}}\left\{s_{2m-1}\right\}\le s_{2m+1}\le s_{2m-1}.\;\;(**)$
From $(*)$ and $(**)$ it follows that
$s_{2m}\le s_{2m+2}\le s\le s_{2m+1}\le s_{2m-1}\;\;\forall m\in\mathbb{N}.$
Therefore
$|s-s_{2m-1}|=s_{2m-1}-s\le s_{2m-1}-s_{2m}=z_{2m}\;\;\forall m\in\mathbb{N}$,
$|s-s_{2m}|=s-s_{2m}\le s_{2m+1}-s_{2m}=z_{2m+1}\;\;\forall m\in\mathbb{N}$.
Hence, in any case it results that
$|s-s_{n}|\le z_{n+1}\;\;\forall n\in\mathbb{N}$.
$$$$REMARK:
In this question the series is convergent for hypothesis, but more generally, if $\left\{z_n\right\}_{n\in\mathbb{N}}$ is not a monotonic sequence, the alternating series $\sum_\limits{n=1}^{\infty}(-1)^{n+1}z_n$ could be not convergent (even though $z_n\to0$ as $n\to\infty$).
For example if $\left\{z_n\right\}_{n\in\mathbb{N}}$ is the following sequence:
$z_{2m-1}=\frac{1}{m}\;$ for all $\;m\in\mathbb{N}$,
$z_{2m}=\frac{1}{m(m+1)}\;$ for all $\;m\in\mathbb{N}$,
the series
$\sum_\limits{n=1}^{\infty}(-1)^{n+1}z_n=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{6}+\frac{1}{3}-\frac{1}{12}+\frac{1}{4}-\frac{1}{20}+\ldots$
is not convergent, indeed
$s_{2p}=\sum_\limits{n=1}^{2p}(-1)^{n+1}z_n=\sum_\limits{m=1}^{p} \left(z_{2m-1}-z_{2m}\right)=\sum_\limits{m=1}^{p} \left(\frac{1}{m}-\frac{1}{m(m+1)}\right)=\sum_\limits{m=1}^{p}\frac{1}{m+1}$
and $\lim_\limits{p\to\infty}s_{2p}=+\infty$.
My remark does not mean that there is not any sequence $\left\{z_n\right\}_{n\in\mathbb{N}}$ non-monotonic for which the alternating series $\sum_\limits{n=1}^\infty (-1)^{n+1}z_n$ is convergent, in fact there are a lot of them.
My remark only means that without the monotony hypothesis, not all the alternating series are convergent, in fact monotony is a sufficient condition for the convergence of the series, but obviously it is not a necessary condition.
