Question 20
Let $m \in \mathbb{N}$, $m \ge 2$ be a fixed natural number, and let $(a_n)_{n \ge 1}$ be a sequence of nonnegative real numbers such that $a_{n+1} \le a_n - a_{mn}$, $\forall n \ge 1$.
a) Prove that the sequence $(b_n)_{n \ge 1}$, $b_n = \sum_{k=1}^{n} a_k$ is bounded above.
b) Prove that the sequence $(c_n)_{n \ge 1}$, $c_n = \sum_{k=1}^{n} k^2 a_k$ is bounded above.
a) Prove that the sequence $(b_n)_{n \ge 1}$, $b_n = \sum_{k=1}^{n} a_k$ is bounded above.
b) Prove that the sequence $(c_n)_{n \ge 1}$, $c_n = \sum_{k=1}^{n} k^2 a_k$ is bounded above.