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## Homework Statement

__Problem:__

Let A_1 , A_2 , . . . be a countable number of finite sets. Prove that the union S = ⋃_i A_i is countable.

__Solution:__

Included in the TheProblemAndSolution.jpg file.

## Homework Equations

Set-theoretic algebra.

## The Attempt at a Solution

Unless I missed something, it seems that I understand everything about that solution except why the author chose to use ##f(b_{ij}) = m_1 + m_2 + . . . + m_{i - 1} + j##.

Could someone please explain to me why that specific function was constructed? I get that the author is trying to show that there is a one-to-one correspondence between the set S and the set of natural numbers, but I don't understand that particular sum was chosen. What's the logic behind the decision?

Any input would be GREATLY appreciated!