Let C be a smooth arc and gk : C → R be continuous for n ∈ N.
a) If gk → g uniformly on C, prove that ∫C gk ds → ∫C g ds as k → ∞.
b) Suppose that {gk} is pointwise monotone and that gk → g pointwise on C as k → ∞. If g is continuous on ɸ(I), prove that ∫C gk ds → ∫C g ds as k → ∞.