Find an increasing sequence of rationals that converges to $\pi$ I am not sure how to construct a sequence that would convey convergence to $\pi$.
Except maybe $a_n=\{\pi + 1/n\}$ but the terms would not be rational.
Looking for an adequate way to show to satisfy the three conditions.

The sequence in graydad's answer can be written in terms of the floor function as
$$n \mapsto 10^{-n} \lfloor 10^n \pi \rfloor.$$
This suggests some cute generalizations. Instead of the decimal expansion for $\pi$, we can use the expansion in any base $q$:
$$n \mapsto q^{-n} \lfloor q^n \pi \rfloor$$
converges to $\pi$ for any whole number $q \ge 2$. In fact,
$$n \mapsto a_n^{-1} \lfloor a_n \pi \rfloor$$
converges to $\pi$ for any increasing sequence $n \mapsto a_n$ of natural numbers!
