Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $f_{n} \to 0$ pointwise on the interval $[-A,A]$ where $f_{n}$ are continuous and uniformly bounded. Can we show the convergence is uniform?

Thanks, any help appreciated.

share|cite|improve this question
Indeed the 'best' similar result in some sense is the Arzela-Ascoli Theorem, which you might be interested to learn. – Edward Hughes Apr 10 '12 at 23:50
This can't not be a duplicate... – t.b. Apr 10 '12 at 23:53
up vote 2 down vote accepted

No. Choose two sequences $(a_n)$ and $(b_n)$ of real numbers so that $0<a_n<b_n<1$ for each $n$ and $\lim\limits_{n\rightarrow \infty} a_n=1$. For each $n$, let $f_n$ be the function on $[0,1]$ whose graph consists of the straight line segments from $(0,0)$ to $(a_n,0)$, from $(a_n,0)$ to $(b_n,1)$, and from $(b_n,1)$ to $(1,0)$.

enter image description here

share|cite|improve this answer

Pointwise convergence does not imply uniform convergence. We have, however, Egorov's theorem, which for the given example states:

Suppose that $f_n$ are measurable and converges almost everywhere to some function $f$ on $[-A,A]$. For every $\epsilon>0$ there exists a measurable set $B_\epsilon\subset [-A,A]$ with measure (total length) $\mu(B_\epsilon) < \epsilon$, such that the convergence is uniform on $[-A,A] \setminus B_\epsilon$.

share|cite|improve this answer

Your Answer


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.