Let f, g be continuous on a closed bounded interval [a, b] with |g(x)| > 0 for x ∈ [a, b]. Suppose that fn → f and gn → g as n → ∞, uniformly on [a, b].
a) Prove that 1/gn is defined for large n and fn/gn → f/g uniformly on [a, b] as n → ∞.
b) Show that a) is false if [a, b] is replaced by (a, b).