I was watching a very-introductory very-basic minicourse (in portuguese, unfortunately) about number theory. In the first minutes of lecture 1, the lecturer (a great one, btw) first defines what is an irrational number: it is a number that cannot be expressed as a quotient of two integers (aka it is not a solution for a polynomial equation of degree 1). Then he starts writing some formulas for $\sqrt 2$ (which would be later proved to be irrational): $$\sqrt 2 = \left(\frac{2\times 2}{1 \times 3}\right)\times\left(\frac{6\times 6}{5 \times 7}\right)\times\left(\frac{10\times 10}{9 \times 11}\right)\times\dots$$
What bugged and I obviously missed something is that this formula says that $\sqrt 2$ is precisely the quotient of two integers. I have no reason to believe that both $2\times2\times6\times6\times10\times10\dots$ is not an integer and the same goes for $1\times3\times4\times5\times9\times11\dots$.
Can anyone please explain what I got wrong about the formula or the theory?
Best Answer
When you see "...", you are looking at an abbreviation for a formal (and sometimes much longer) statement. What the formula means is that if $P(n)=\prod_{j=1}^n\frac {(4j-2)^2}{(4j-3)(4j-1)}$ whenever $n\in \Bbb N,$ then $\lim_{n\to\infty}P(n)=\sqrt 2.$
And $\lim_{n\to \infty}P(n)=\sqrt 2$ is itself an abbreviation for $\forall r>0\,\exists n_r\in \Bbb N\, \forall n\in \Bbb N\, (n\ge n_r\implies |P(n)-\sqrt 2|<r).$