18.099b Problem Set 3a Due: Thursday, Match 3rd (in class or before).

Suppose X ⊆ ℝ is such that every infinite subset of X contains an infinite convergent sequence. That is, if F ⊆ X is infinite then there exists an infinite sequence {a_{n}} with a_{n} ∈ F for all n, and a number a ∈ ℝ, such that lim_{n→∞} a_{n} = a.

Prove that X is bounded from above and below.

Hint: Assume, for a contradiction, that the conclusion fails. Construct an infinite set {p_{1}, p_{2}, … } in X such that for any i ≠ j |p_{i} − p_{j}| ≥ 1. Use this to contradict our assumption on X.