HMMT 2011 T9

We would now like to examine the behavior of $p_{m}(k)$ as $m$ becomes arbitrarily large; specifically, we would like to discern whether $\lim _{m \rightarrow \infty} p_{m}(0)$ exists and, if it does, to determine its value. Let $\lim _{m \rightarrow \infty} p_{m}(k)=A_{k}$. (a) [5] Prove that $\frac{2}{3} p_{m}(k) \geq p_{m}(k+2)$ for all $m$ and $k$. (b) [10] Prove that $A_{0}$ exists and that $A_{0}>0$. Feel free to assume the result of analysis that a non-increasing sequence of real numbers that is bounded below by a constant converges to a limit that is greater than or equal to $c$.