MathNet 2025 — Part 02 (100 complete bilingual questions) Question 7

Let $A_n$ denote the number of subsets of $\{1, 2, \dots, n\}$ that contain no two elements differing by $1$, and let $B_n$ denote the number of subsets that contain no two elements differing by $2$. Prove that $A_{2025} \lt B_{1013}^2$.
(Batbayasgalan Balkhuu and Nursoltan Khavalbolot)