## Dini’s Theorem

Let be two real numbers and a sequence of continuous functions which converge pointwise to a continuous function .

**Dini’s Theorem.**Suppose that the sequence has the property that . Prove that the convergence is uniform.- Suppose that every function is increasing. Prove that the convergence is uniform.
- If every function is convex on , prove that the convergence is uniform on every compact interval in .

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Categories: Analysis
ecole-polytechnique

I don’t understand the third problem. convergence is uniform on every compact in . what does that mean

It is a typo. T should have written compact interval…

It means that the convergence may not be uniform on , but for any the convergence is uniform.

Dinis theorem needs the continuity of !! See over , for example.

You are right. I corrected the statement of the problem.