Question: Suppose that fn f and gn g, as n , uniformly on some set E R. a) Prove that fn +

Suppose that fn → f and gn → g, as n → ∞, uniformly on some set E ⊂ R.
a) Prove that fn + gn → f + g and αfn → αf → αf, as n → ∞ uniformly on E for all α ∈ R.
b) Prove that fngn → fg pointwise on E.
c) Prove that if f and g are bounded on E, then fngn on E. then fngn → fg uniformly on E.
d) Show that c) may be false when g is unbounded.

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