I am reading this wikipedea article on the proof of irrationality of $\sqrt{2}$. It uses the principle of *infinite descent*. I understand it as:

- We assume $\sqrt{2}=\dfrac pq$, where $p$ and $q$ are some positive integers.
- We have $2q^2 = p^2 \implies$ $p$ is even, i.e. $p=2r$ for some positive integer $r$.
- Now we have $2q^2=(2r)^2 \implies q$ is also even, i.e. $q=2s$ for some positive integer $s$.

From steps 1,2 and 3 we conclude that both $p$ and $q$ have $2$ as their factor *at least* one time. So we can say $\sqrt{2}=\dfrac pq = \dfrac rs$.

Now we can repeat the steps 1,2 and 3 with $\sqrt{2}= \dfrac rs$. This will eventually imply that intezers $r$ and $s$ also have 2 as their factor at least one time, or $r=2r_1$ and $s=2s_2$. Recalling $p=2r$ and $q=2s$ implies that $p$ and $q$ have $2$ as their factor *at least* two number of times.

The notable fact is that we can repeat steps 1,2 and 3 infinite number times, which implies that $p$ and $q$ have 2 as their factor infinite number of times, but this can't be for any finite $p$ and $q$, that is to say their does not exist any intezer which can have 2 as its factor infinite number of times. So we conclude that there doesn't exist any $p$ and $q$ which satisfies $\sqrt 2 = \dfrac pq$, hence $\sqrt 2$ is irrational.

**Question 1:** Do I understand the proof correctly?

**Question 2:** I noted that this proof on wikipedea is titled as *Proof by infinite descent*, where it does not use the concept of infinite descent at all. So why is it titled as *Proof by infinite descent*?