Sequences and Series of Functions

The following theorem is an immediate consequence of 2.1.11 and 3.1.9.

7.1.2 Theorem. Let fn → f and gn → g pointwise on S and let h be continuous such that h ◦ fn and h ◦ f are defined on S. Then, for α, β ∈ R,

αfn + βgn → αf + βg, fngn → fg, fn gn → f

g (if g 6= 0) and h ◦ fn → h ◦ f

pointwise on S.