For each , let fn() be a real-valued sequence. We say fn() converges uniformly (in

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

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

θ∈

| fn(θ) − f (θ)| → 0 as n → ∞. If 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 is countable.

Section 11.2