Problem

Source: 2022 Taiwan TST Round 1 Independent Study 2-A

Tags: inequalities, Taiwan



Let $a_1, a_2, a_3, \ldots$ be a sequence of reals such that there exists $N\in\mathbb{N}$ so that $a_n=1$ for all $n\geq N$, and for all $n\geq 2$ we have \[a_{n}\leq a_{n-1}+2^{-n}a_{2n}.\]Show that $a_k>1-2^{-k}$ for all $k\in\mathbb{N}$. Proposed by usjl