Question 92
A sequence of real numbers $a_1, a_2, \dots$ has the following properties:
i. there is a positive integer $N$ such that $a_n = 1$ for all $n \ge N$;
ii. for all $n \ge 2$,
$$
a_n \le a_{n-1} + 2^{-n} a_{2n}.
$$
Prove that $a_n \gt 1 - 2^{-n}$ for all positive integers $n$.
i. there is a positive integer $N$ such that $a_n = 1$ for all $n \ge N$;
ii. for all $n \ge 2$,
$$
a_n \le a_{n-1} + 2^{-n} a_{2n}.
$$
Prove that $a_n \gt 1 - 2^{-n}$ for all positive integers $n$.