Let us assume that there exist a smallest positive integer that is neither odd nor even, say n. Since n is least positive integer which is neither even nor odd, n – 1 must be either odd or even.
Case 1: If n – 1 is even, n – 1 = 2k for some k.
But this implies n = 2k + 1 this implies n is odd.
Case 2: If n – 1 is odd, n – 1 = 2k + 1 for some k.
But this implies n = 2k + 2 (k+1) this implies n is even.
In both ways we have a contradiction.
Thus, every positive integer is either even or odd