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## Dini’s Theorem

Let $a be two real numbers and $f_n :[a,b ]\to \Bbb{R}$ a sequence of  continuous functions which converge pointwise to a continuous function $f$.

1. Dini’s Theorem. Suppose that the sequence $(f_n)$ has the property that $f_n \leq f_{n+1}$. Prove that the convergence $f_n \to f$ is uniform.
2. Suppose that every function $f_n$ is increasing. Prove that the convergence $f_n \to f$ is uniform.
3. If every function $f_n$ is convex on $[a,b]$, prove that the convergence $f_n \to f$ is uniform on every compact interval in $(a,b)$.
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1. February 24, 2011 at 8:07 am

I don’t understand the third problem. convergence $f_{n} \to f$ is uniform on every compact in $(a,b)$. what does that mean

• February 24, 2011 at 10:41 am

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

It means that the convergence may not be uniform on $[a,b]$, but for any $[c,d] \subset [a,b]$ the convergence is uniform.

2. March 11, 2011 at 2:41 pm

Dinis theorem needs the continuity of $f$ !! See $f_n(x)=x^n$ over $[0,1]$, for example.

• March 11, 2011 at 8:24 pm

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