Let $A_k$ be the number of pairs $(a_1, a_2, ..., a_{2k})$ for $k\leq 50$, where $a_1, a_2, ..., a_{2k}$ are all different positive integers that satisfy the following. $\cdot$ $a_1, a_2, ..., a_{2k} \leq 100$ $\cdot$ For an odd number less or equal than $2k-1$, we have $a_i > a_{i+1}$ $\cdot$ For an even number less or equal than $2k-2$, we have $a_i < a_{i+1}$ Prove that $A_1 \leq A_2 \leq \cdots \leq A_{49}$.