## Proof that irrational numbers do not exist

I understand exactly what you are saying. But you haven't added anything new the mathematics. It is, in fact, well known that every rational number is the limit of a sequence of irrational numbers. This is pretty much what you have shown. It is not a surprise to anyone who has taken a semester of Real Analysis.

The fact that there are reals that cannot be written in terms of integer ratios is, in fact, an important distinction if only because of the fact that they are countable whereas the entire real line isn't. There is nothing trivial about that.

Putting this aside, I grant that what you are saying is, in some sense, correct (at least what you wrote in the OP is "correct", even if it is trivial). So, what is the result you have proven with respect to prime numbers?

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 Quote by Robert1986 I understand exactly what you are saying. But you haven't added anything new the mathematics. It is, in fact, well known that every rational number is the limit of a sequence of irrational numbers. This is pretty much what you have shown. It is not a surprise to anyone who has taken a semester of Real Analysis. The fact that there are reals that cannot be written in terms of integer ratios is, in fact, an important distinction if only because of the fact that they are countable whereas the entire real line isn't. There is nothing trivial about that. Putting this aside, I grant that what you are saying is, in some sense, correct (at least what you wrote in the OP is "correct", even if it is trivial). So, what is the result you have proven with respect to prime numbers?
I dare say that what the OP purports to prove is false because all rational numbers are either finite decimal numbers or infinite repeating decimals. That is only some infinite decimal representations, e.g. 12.66123123123..., where the ending part, i.e. 123, repeats forever are rational numbers while those infinite decimal numbers which do not have an repeating ending are irrational numbers.

 Quote by ramsey2879 I dare say that what the OP purports to prove is false because all rational numbers are either finite decimal numbers or infinite repeating decimals. That is only some infinite decimal representations, e.g. 12.66123123123..., where the ending part, i.e. 123, repeats forever are rational numbers while those infinite decimal numbers which do not have an repeating ending are irrational numbers.
When I said that his OP was "correct" I only meant that he was correct in asserting that irrational numbers are the limits of sequences of rational numbers. But, as I said, that is not new to anyone who was taken a course in Real Analysis.

I am just interested in what he has allegedly proven about prime numbers.

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