Tests For Uniform Convergence
Tests For Uniform Convergence : Theorem 1 (Mₙ _ Test) : Let <fₙ> be a sequence of function defined on a metric space X . Let lim fₙ(x) = f(x) ∀ x ∈ X and let n-->∞ Mₙ = Sup {|fₙ(x) - f(x)| : x∈ X} Then <fₙ> converges uniformly to f iff Mₙ-->0 as n-->∞. Proof Of Necessary Part : Let us suppose the sequence <fₙ> of functions converges uniformly to f on X . Then by definition , for a given ε > 0 ∃ a positive integer m (independent of x) such that n≥ m ⇒|fₙ(x) - f(x)| < ε ∀ x∈X Also , Mₙ is the supremum of |fₙ(x) - f(x)|. Therefore |fₙ(x) - f(x)| < ε ∀ n≥m ∀ x∈X ⇒ Mₙ = Sup |fₙ(x) - f(x)| < ε ∀ n≥ m x∈X ⇒ |fₙ(x) - f(x)| ≤ Mₙ <ε ∀ n≥m ,∀ x∈X ⇒ <fₙ> converges uniformly to f on X . For Example : Show that the sequence <fₙ(x)> where fₙ