Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $f_n: [a,b] \rightarrow, n \in \mathbb{N}$, be a sequence of functions converging uniformly to $f: [a,b] \rightarrow \mathbb{R}$ on $[a,b]$. Suppose that each $f_n$ is continuous on [a,b] and differentiable on (a,b), and that the sequence of derivatives $(f'_n)$ is uniformly bounded on (a,b). This means that there exists an $M>0$ such that $|f'_n(x)| \le M$ for all $x \in (a,b)$ and all $n \in \mathbb{N}$

Question: Show that $(f_n)$ is equicontinuous.

Known definitions:

  • A sequence of functions $(f_n)$ converges uniformly to a limit function $f$ on a set $A$, if, for every $\epsilon >0$ , there exists an $N\in \mathbb{N}$. such that$|f_n(x) - f(x)| < \epsilon$ whenever $n \ge N$ and $x \in A$
  • Cauchy Criterion for Uniform Convergence: A sequence of functions $(f_n)$ converges uniformly on a set $A$, if and only if, for every $\epsilon >0$ , there exists an $N\in \mathbb{N}$. such that$|f_n(x) - f_m(x)| < \epsilon$ for all $n,m \ge N$ and all $x \in A$
  • A sequence of functions $(f_n)$ defined on a set $E$, is called equicontinuous if for every $\epsilon >0$ , there exists a $\delta>0$ such that $N\in \mathbb{N}$. such that$|f_n(x) - f_n(y)| < \epsilon$ for all $n \in N$ and $|x-y| \lt \delta in E$
  • A sequence of derivatives $(f′n)$ is uniformly bounded on (a,b) if there exists an $M>0$ such that $|f′n(x)|≤M$ for all $x∈(a,b)$ and all $n∈N$
share|improve this question

2 Answers 2

up vote 2 down vote accepted

You only need the uniform boundedness of the derivatives. Let $\epsilon > 0$, and choose $\delta = \epsilon/M$. Then for all $x<y \in [a,b]$ such that $|x-y| < \delta$, for every $n$, by the mean value theorem, there exists $c_n \in (x,y)$ such that $f_n(y) - f_n(x) = f_n'(c_n) \cdot (y - x)$. Therefore:

$$|f_n(y)-f_n(x)| \leq |f_n'(c_n)| \cdot |x - y| < M \cdot \epsilon/M = \epsilon$$

share|improve this answer

Hint

$$ |f_n(x)-f_n(y)|=|(f_n(x)-f(x))+(f(x)-f(y))+(f(y)-f_n(y))| $$

$$ \leq |f_n(x)-f(x)|+|f(x)-f(y)|+|f(y)-f_n(y)| \,.$$

Now, use the assumptions you have been given.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.