Proof of irrationality of sqrt2 in Penroses Road to Reality

[Feymar]

Hello Everybody, I hope that I've picked the right sub-forum for my question.

The problem is as follows:

I understand perfectly well the proof by contradiction of the irrationality of the square root of 2, where we prove that if we assume that it is rational, then the integers of which the ratio is made of can't be coprime, which is a condratiction.

Well, but recently I came along this amazing book of Roger Penrose "The Road to Reality". I love it and most of it I can understand or manage to find some more elaborative explanation somewhere. But there was this one issue, to which I keep coming back, seemingly one of the simplest ones but that I just fail to grasp.

In the book he mentions the proof by contradiction of the irrationality of the square root of 2, and outlines the proof, but by the end instead of just saying that the contradiction lies in the impossibility of the two integers to be coprimes, he writes as follows:

"But any decreasing sequence of positive integers must come to an end, contradicting the fact that this sequence is unending. This provides us with a contradiction to what has been supposed, namely that there is a rational number which squares to 2."

I know that I did not write the full proof here, up until that point it is the same as outlined in any textbook and any page you come across online, so I believe that there is no need for it. If it is, I will gladly write the full proof as outlined in the book here.

Now, the thing that I do not understand is what has an unending sequence to do with all of this? Where did that come from and how does it make sense that if (hipothetically) there was an unending sequence, there would have been a rational number which squares to 2 , or what?

I just don't see the connection that he made all of a sudden between finite and infinite sequences and the (non)existence of a rational sqrt2. He did not just merely wrote something along the lines of: "And since we get that both integers must be even if we assume that the sqrt of 2 is a rational number, we also get that the integers can't be coprimes which is a contradiction to the assumption that the sqrt of 2 is rational." or something like that.

No, he wrote what he wrote, and I can't quite comprehend the connection between the proof and how he , through the proof concluded what is written in the quote. If somebody could help me with this I would highly appreciate it.

Sorry that I made it so long, I must confess that I am not a man of few words, you'll have to forgive me for that.

 

[mfb]

I don't have the book, but it sounds like an unnecessary complication.

Instead of starting with a/b without common factors, you can start with a/b where you accept common factors, and then show that you can reduce this to (a/2)/(b/2) and so on until you get a contradiction.

 

[mathman]

Each step in the proof leads to a ratio of numbers not co-prime and smaller tan the previous pair. Repeating this process leads to a stop since the number of positive integers less than the previous pair is finite. However the proof says it should never stop.