Consider a set S of pairs of integers. The set is defined as follows. Pair (0, 0) is in S, i.e. (0, 0) ∈ S. If some pair (a, b) ∈ S, then the following pairs are also in S: (a, b + 1) ∈ S, (a + 1, b + 1) ∈ S, and (a + 2, b + 1) ∈ S. Prove (by induction) that for each pair (a, b) ∈ S we have that a ≤ 2b. Hint: observe that S is defined recursively, its first four elements are S = {(0, 0),(0, 1),(1, 1),(2, 1), . . .}.