Prove by induction that for all natural numbers, n, either 3|n or 3|n+1 or 3|n+2? That is prove that for all natural numbers, n, either 3 is a factor of n or n+1 or n+2

Base case: n = 1.
Then 3 divides 3, which is n+2.
Inductive hypothesis: 
if the proposition is true for n: Then proof by cases
1). If n is divisible by 3, this implies that (n+1) + 2 is also divisible by 3.
2). If n +1 is divisible by 3, then (n+1) is also divisible by 3.
3). If n+2 is divisible by 3, then (n+1) + 1 is also divisible by 3.
Thus, for all natural numbers ,n, the 3 divides either n, n+1, or n+2.
