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2 of 8 showed the non-decreasing claim

"this class of problems lie"s in RE, so its name is "RE".




$\operatorname{Prob}(\hspace{.02 in}M \text{ accepts}) \; = \; \operatorname{Prob}\hspace{-0.04 in}\big(\hspace{-0.02 in}(\exists n)(\hspace{.02 in}M \text{ accepts after exactly } n \text{ steps})\big)$
$=$

$\displaystyle\sum_n \operatorname{Prob}(\hspace{.02 in}M \text{ accepts after exactly } n \text{ steps})$

$=$
$\displaystyle\lim_N \left(\displaystyle\sum_{n\leq \hspace{.02 in}N} \operatorname{Prob}(\hspace{.02 in}M \text{ accepts after exactly } n \text{ steps})\hspace{-0.04 in}\right)$

$=$

$\displaystyle\lim_N \: \operatorname{Prob}(\hspace{.02 in}M \text{ accepts in at most } N \text{ steps}) \;\;\; = \;\;\; \displaystyle\lim_n \: \operatorname{Prob}(\hspace{.02 in}M \text{ accepts in at most } n \text{ steps})$



For all $m$ and $n$, if $\: m\leq n \:$ then for all randomness strings $r$,

$\text{on randomness } r, \: M \text{ accepts in at most } m \text{ steps}$
$\iff$
$(\exists t)\big((\text{on randomness } r, \: M \text{ accepts after exactly } t \text{ steps}) \: \text{ and } \: t\leq m\big)$
$\implies$
$(\exists t)\big((\text{on randomness } r, \: M \text{ accepts after exactly } t \text{ steps}) \: \text{ and } \: t\leq n\big)$
$\iff$
$\text{on randomness } r, \: M \text{ accepts in at most } n \text{ steps}$

.


For all $m$ and $n$, if $\: m\leq n \:$ then $\operatorname{Prob}(\hspace{.02 in}M \text{ accepts in at most } m \text{ steps}) \; \leq \; \operatorname{Prob}(\hspace{.02 in}M \text{ accepts in at most } n \text{ steps}) \;\;\;\;$.



$\frac12 \;\; < \;\; \operatorname{Prob}(\hspace{.02 in}M \text{ accepts})$

$\iff$

$\frac12 \;\; < \;\; \displaystyle\lim_n \: \operatorname{Prob}(\hspace{.02 in}M \text{ accepts in at most } n \text{ steps})$

$\iff$

$(\exists n)\left(\frac12 < \operatorname{Prob}(\hspace{.02 in}M \text{ accepts in at most } n \text{ steps})\hspace{-0.02 in}\right)$