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.

Let $n_0:=\lfloor\pi\rfloor$ and define inductively $$n_{k+1}:=\left\lfloor10^{k+1}\left(\pi-\sum_{i=0}^k\dfrac{n_i}{10^i}\right)\right\rfloor.$$ Then $$\lim_{m\to\infty}\sum_{k=0}^m\dfrac{n_k}{10^k}=\sum_{k=0}^\infty\dfrac{n_k}{10^k}=\pi.$$
