I am confused about the following statement in someone's paper:
Since the probability simplex is compact, the sequence $\{a^{(T)}_n\}_{n\in [d]}$ belonging to the probability simplex for any $T$ converges to a vector $w$ in the probability simplex when $T\to\infty$. $T$ is an index and we assume it is any integer greater than 1. $d$ is the dimension of the probability simplex.
You can image $\{a^{(T)}_n\}_{n\in [d]}=[a^{(T)}_1, a^{(T)}_2, ..., a^{(T)}_d]$ is a probability vector but each element may change over time $T$.
However, let's consider 2-dimension probability simplex, I am confused about whether the following sequence would converge to a vector in probability simplex or not:
If $\{a^{(1)}_n\}_{n\in [2]}$ = [1, 0], $\{a^{(1)}_n\}_{n\in [2]}$ = [0, 1], $\{a^{(3)}_n\}_{n\in [2]}$ = [1, 0], ... and so on. That is, when T is even, $\{a^{(T)}_n\}_{n\in [2]} = [0, 1]$ but when T is odd, $\{a^{(T)}_n\}_{n\in [2]}$ = [1, 0].
So for any $T\geq 1$, $\{a^{(T)}_n\}_{n\in [2]}$ is a probaiblity vector, but can we say $\{a^{(T)}_n\}_{n\in [2]}$ converges to a vector $w$ in the probability simplex? can we say $\lim_{T\to\infty} \{a^{(T)}_n\}_{n\in [2]} = w$? If the answer is yes, is it related to the compactness of the probability simplex?