Consider a sequence bn (containing infinite amount of values) with values in ℝ, indexed by ℕ. Suppose the range of bn is contained in the finite interval [a, b].
Use the Principle of Mathematical Induction to prove that for any n ∈ ℕ, there is an interval of length (b-a)⁄2n, k ≥ 1, which contains infinitely many terms of the sequence
I can understand it intuitively, bn has infinite terms, so there exists a subset of bn that also has infinite terms. However, I am not sure how I would go about predicting this inductively.