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A proof is literally "proof" that's the point, you can't prove it, it is proof.
n+1 = odd if n is even and even if n is odd.
Prove it because of the definition of even and odd.
Even is divisible by 2 and odd is not divisible by 2 according to the definition for this purpose.
A number that is divisible by 2 needs to have a factor of 2.
A number with a factor of 2 (even) + 1 will no longer have a factor of 2 because 1 put it at an offset that is not sufficient to be divided by 2 evenly because the extra 1 will convert to 1/2.
Vice versa happens with an odd, which will always have a hanging 1 if it is an integer so odd + 1 = even.
n+1 is proven the equation that converts odds to evens and evens to odds.
If you multiply all primes before a given prime and add 1 that new number must have a prime factor that is not any of the prime numbers you used because the 1 put it at an offset. That is proof there are infinite prime numbers too.
