18.099b Problem Set 2a Due: Thursday, February 26th (in class or before).

1. Suppose F ⊆ ℝ is a subset of the real numbers for which sup F exists. Show that sup F = lim_{n→∞} a_{n} for some sequence {a_{n}} of numbers in F. (Note: sup F is the least upper bound of F: the least number α such that x ≤ α for all x ∈ F.)

2. Let {a_{n}} be a sequence of real numbers. Prove that if {a_{n}} converges than so does {|a_{n}|}.