EDB — 1JV

Exercises

1. [1JV]Let \(I⊂ℝ\) be a compact interval, let \(f_ n,f:I→ℝ\) be continuous. Show that the following two facts are equivalent.

   a.

   For every \(x∈ X\) and for every sequence \((x_ n)_ n⊂ I\) for which \(x_ n→_ n x\), we have \(\lim _{n→∞} f_ n(x_ n)=f(x)\);

   b.

   \(f_ n→_ nf\) uniformly on \(I\).

   Then find an example where \(I=[0,1)\), the first point holds, but \(f_ n\) does not tend uniformly to \(f\).

   Solution 1

   [1JW]↺↻