First-Consider a number (N) with finite number of digits say (M), of string (999...), this of course can be factored into prime numbers only
Second-Now add one to the previous number (N), (999..+1), you will get another number (N+1) with COMPLETELY deferent prime factorization
Third repeat the process for (M+1) digits, and this is the principle of Induction method of the proof, where you would find that always applicable, then
You will always get two sets of prime factorization for (N), and (N+1), where can never be considered exactly equal, there fore their division (N/(N+1)) can not be equal to one EXACTLY, except by consideration or limit or convention
Is not this is a rigorous proof for such a SILLY problem?
So, most you do accept that (N = N+1), when (N) is sufficiently large or becomes Infinite despite having totally deferent prime factorization, that is to say, and of course primes are infinite
Gcd (N, N+1) = 1
But the illusion in the number system it self One may ask how?
I will tell you HOW?
How do you now that ONE is real number? It is only by CONVENTION, isn't it?
Other wise in the binary system it becomes like
.1111..., is also one, and similarly in base three 0.222... is also one, and so on to any system number, which will lead you to many contradiction in your real life system
However my definition shows you that all those endless (not all zeros) digits numbers are IRRATIONAL numbers when ONLY expressed in endless digital system but NOT as fractions, other wise provide me with only ONE counter example to my real number system
You should also note that there are ONLY FIVE NUMBERS that aren't REAL, EXCEPT by CONVENTIONS, those are (ZERO, +/- ONE, +/- INFINITY), in my real number system
I guess you will not accept that also..,
May be because you are the same inhabitants of the old centuries, NOTHING really changed...!!
And I wonder how to prove a much larger facts for you people!! .