Consider the set of natural numbers, N. For any given natural number, n, every n-th natural number is a multiple of n, hence 1/n of N are multiples of n.
In the set of even numbers, E, every n-th number is also a multiple of n, hence 1/n of E are multiples of n.
We can generalize this statement that for the set of all multiples of any given prime number, Pn, and we will call this set Mn, every n-th member of this set is a multiple of n, hence 1/n of Mn are multiples of n.
If we remove E from N, and are left with with the set of odd numbers, O, then for any given n, 1/n of O are multiples of n.
Consider now the process of stepwise removal of all multiples of all primes, Pn, working from 2 toward infinity. Removal of the even numbers leaves 1/2 of N remaining under consideration as possible prime numbers.
Of the remaining half of N, 1/3 are multiples of 3, so further removal of composite numbers to eventually reveal the set of primes removes 1/3 of the remaining half of N, or 1/3 of 1/2, leaving a rational number, 1/3, of N remaining under consideration as possible primes.
When we next remove the remaining products of 5, we strip away 1/5 of what was left prior, or 1/5 of 1/3, or 1/15, leaving 8/30 of N.
In progressive steps of sieving with each successive prime, we strip away a fraction of the candidate primes remaining after the prior sieving step, where this fraction is in all cases a rational number, this leaving a rational number as the fraction of N remaining as possible primes.
As this process proceeds toward infinity, the fraction of N remaining with candidate primes gets ever smaller, approaching but never reaching zero, remaining always a rational number. Since N is an infinite set, any rational fraction of N is also infinite. Hence the set of prime numbers is infinite.
We can use similar logic to show that the sets of twin primes and larger prime k-tuples are also infinite.
Consider again stripping away all even numbers, leaving O. At this stage, every remaining number, which remains to this point under consideration as a possible prime, is separated from another candidate prime by two, hence every odd number is a candidate twin prime, and half of N remain as possible twin primes. Let’s call the set of candidate twins after n sieving steps as TWn.
Twin prime pairs (the “normal” definition) are groups of two consecutive primes separated by two. Consider that every twin pair, or every candidate pair remaining after the n-th sieving stage, has a lower prime and an upper prime. Define the set of lower primes within the n-th sieving stage set of candidate twins as T_L and the set of upper primes as T_U. Within the set of candidate twins, half are in T_L and half are in T_U, and T_L and T_U are offset by two, with all members of T_U being equal to the corresponding member of T_L plus 2. We can therefore deduce that within both T_L and T_U, 1/Pj of each set are multiples of Pj where Pj are primes higher than Pn.
When we remove all primes P(n+1) in the next stage of sieving, we remove 1/P(n+1) from T_L and 1/P(n+1) from T_U. Every member of T_L or T_U thus removed eliminates one candidate twin pair from further consideration, and since any single repetition of P(n+1) cannot eliminate both primes in a single pair, 1/P(n+1) of TWn are removed from both T_L and T_U, or 2/P(n+1) of TWn is eliminated.
Thus we see that, while 1/Pn of all remaining prime candidates are removed at each sieving stage, 2/Pn of all remaining twin prime pair candidates are removed. And yet, following a similar logic as demonstrated for the sequential elimination of prime candidates, the proportion of N remaining as candidate twin primes is always a rational number, no matter how arbitrarily large Pn becomes. Hence, there are infinitely many twin primes.
Following precisely the same logic, we can show that for Pn > k, where k is the size of a prime k-tuple, each sieving stage removes k/Pn of the set of candidate k-tuples remaining after the previous sieving stage, and hence the sets of all k-tuples for finite k, where feasible k-tuples exist, are infinite.