EDB — 1HS

Exercises

1. [1HS] Note:This result is known as ”Dini’s lemma”.
   Let \((X,d)\) be a metric space, let \(I⊂ X\) be a compact set, and suppose that \(f,f_ n:I→ℝ\) are continuous and such that \(f_ n(x)↘_ n f(x)\) pointwise (i.e. for every \(x∈ I\) and \(n\) we have \(f(x)≤ f_{n+1}(x) ≤ f_{n}(x)\) and \(\lim _ n f_ n(x) =f(x)\)). Show that \(f_ n→ f\) uniformly.

   Solution 1

   [1HT]↺↻

   Solution 2

   [1HV]↺↻

   In following exercises we will see that, if even one of the hypotheses fails, then there are counterexamples.