I was solving this problem that needed to guess some expected value identity. I tried to prove it but couldn't.

There are $$$n$$$ pairs of numbers $$$(a,b)$$$. In one operation, you will choose one pair uniformly randomly. Let $$$(X_i,Y_i)$$$ denote the chosen pair in the $$$i^{th}$$$ operation.

Prove that $$$\large{E(\frac{\sum_{i=1}^{k}X_i}{\sum_{i=1}^{k}Y_i}) == \frac{E(\sum_{i=1}^{k}X_i)}{E(\sum_{i=1}^{k}Y_i)} == \frac{\sum_{j=1}^{n}a_j}{\sum_{j=1}^{n}b_j}}$$$ as $$$k$$$ tends to infinity.