First Part :I want to prove that, if I have a sequence of independent Randomvariables $X_{n}$ and $T=\{\exists N \in \mathbb{N}:\forall n\geq N, X_{n}=X_{n+2}\}$, that $P(T)\in\{0,1\}$. I know that I have to show that T is in the Terminal sigma Algebra and then use Kolmogrovs Zero One Law. But I dont know how to show that T is in the terminal sigma Algebra.
Second Part: If $P(T)=1$ and the sequence is i.i.d, then $X_{i}$ is almost sure constant. How would I proof this ?
Define $\mathcal{F}_n:=\sigma(X_n,X_{n+1},\ldots)$. Then the tail $\sigma$-algebra is $\tau=\bigcap_{n\ge 1}\mathcal{F}_n$. It is clear that the event $T$ belongs to each $\mathcal{F}_n$ because it does not depend on $(X_1,X_2,\ldots, X_m)$ for any finite $m$, i.e., $$ T=\{\exists N: \forall n\ge N\vee m, X_{n}=X_{n+2}\}\in \mathcal{F}_m $$ for any $m\ge 1$. Thus, $T\in \tau$ and by Kolmogorov's 0-1 law, $\mathsf{P}(T)\in \{0,1\}$.
Next, if $\mathsf{P}(T)=1$, then $$ \lim_{n\to\infty}\mathsf{P}\!\left(\bigcap_{m\ge n}\{X_m=X_{m+2}\}\right)=1. $$ However, if $X_1$ is not a constant, \begin{align} \mathsf{P}\!\left(\bigcap_{m\ge n}\{X_m=X_{m+2}\}\right)&\le \mathsf{P}\!\left(\bigcap_{m\ge n}\{X_{3m}=X_{3m+2}\}\right) \\ &=\prod_{m\ge n}\mathsf{P}(X_{3m}=X_{3m+2})=0. \end{align}