For each , let fn() be a real-valued sequence. We say fn() converges uniformly (in ) to f() if sup |fn()

For each θ ∈ Ω, let fn(θ) be a real-valued sequence. We say fn(θ)

converges uniformly (in θ) to f(θ) if sup

θ∈Ω

|fn(θ) − f(θ)| → 0 as n → ∞. If Ω if a finite set, show that the pointwise convergence fn(θ) → f(θ)

for each fixed θ implies uniform convergence. However, show the converse can fail even if Ω is countable.

Section 11.2