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 $X$ be a metric space and $\{f_n\}$ be a sequence of functions such that $f_n:E\rightarrow X$.

Suppose $f_n\rightarrow f$ uniformly on a set $E$ and $x$ is a limit point of $E$ and $\lim_{t\to x} f_n(t)=A_n$. ($E$ is a subset of some metric space $Y$.)

I know that if $X$ is complete, then $\{A_n\}$ converges. However, i'm really not sure if it is essential. Is it?

share|cite|improve this question
Do you want $E$ to be a subset of some larger space? Or do you want uniform convergence of the $f_n$ on some subset of $E$? – David Mitra Dec 13 '12 at 12:57
@David I meant $E$ to be a subset of arbitrary metric space – Katlus Dec 13 '12 at 13:10
up vote 2 down vote accepted

Without the completeness assumption, the statement is false. To see this, take $X=[0,1)$, $Y=[0,1]$, $E=[0,1)$, and for a positive integer $n$, define $f_n:E\rightarrow X$ by $f_n(x) ={n-1\over n} x $. Here you have, for the limit point $1$ of $E$, that $A_n={n-1\over n}$. Now note the sequence $\{A_n\}$ does not converge in $X$.

share|cite|improve this answer
Best! Thank you :) – Katlus Dec 13 '12 at 13:25
@Katlus You're welcome. – David Mitra Dec 13 '12 at 13:28

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.