In an exercise, if $ \{ X_n \} $ is a random variable sequence and $X$ another random variable, I am requested to show that the event sequence $\{T_n\}$ is increasing, where:
$$ T_m = \{ |X_n-X| < ε \quad \forall \; n \geq m \} \quad , m \in \mathbb{N} $$ for some $ε>0$.
However I think that it is decreasing, because $$ T_m = \{ |X_n-X| < ε \quad \forall \; n \geq m \} \supset \{ |X_n-X| < ε \quad \forall \; n \geq m+1 \} = T_{m+1} $$
Where is the mistake?
Thank you in advance
On
I think the most important thing here is to realize the condition $\textbf{ for all } n \geq m$, which can be a very small ('strict') event, e.g. 'all tosses=Heads after $m^{th}$ toss', so if you define an indicator function for the event: $$ I_{T_{m}} = \mathbf{1}_{|X_{n \geq m}-X| < \varepsilon} $$ This event will be all $1s$ for $n \geq m$. On the other hand for $m+1, I_{T_{m+1}}$ includes both $I_{T_{m}} = \{0,1\}$, i.e. both the event and its compliment, so it's a larger (less strict) event, hence $I_{T_{m}} \subset I_{T_{m+1}} $ and, hence $ T_{m} \subset T_{m+1}$
The inclusion is actually wrong.
Let $\omega\in T_m$. Then, by definition: $$|X_n(\omega)-X(\omega)|<\varepsilon \quad \forall n\ge m.$$ This holds for all $n\ge m$, then holds in particular for all $n\ge m+1$ and hence $$\omega\in T_{m+1},$$ so you have $$T_m \subseteq T_{m+1}$$
You got confused in the inclusion: if $\omega\in T_{m+1}$ than you don't know anything about $n=m$.