How to solve the following task:

Show that if $f_n$ is a sequence of uniformly converging mappings $f_n \in C[0,1]$, where $C[0,1]=\{f:[0,1]\rightarrow\mathbb{R} \;\mid\; f\; \text{continuous}\}$ then

$\displaystyle\underset{n\rightarrow \infty}{\lim}\int_0^1 f_n = \int_0^1\underset{n\rightarrow \infty}{\lim} f_n$.

How to tackle this problem? Should I consider using some metric function here or?

  • $\begingroup$ Hi @ClementC. you're right. Let me double check my problem statement. $\endgroup$ – jjepsuomi Feb 9 '16 at 20:58
  • $\begingroup$ I edited the question. $\endgroup$ – jjepsuomi Feb 9 '16 at 21:02

Write $$ \left\lvert \int_0^1 f_n - \int_0^1 f\right\rvert \leq \int_0^1 \left\lvert f_n - f\right\rvert \leq \int_0^1 \lVert f_n - f\rVert_\infty = \lVert f_n - f\rVert_\infty. $$ What does uniform convergence give you then?

  • $\begingroup$ Hi @ClementC I think you got it right. I had a confusion with the notation in my problem statement I think. Thank you! =) $\endgroup$ – jjepsuomi Feb 9 '16 at 21:04
  • $\begingroup$ You're welcome. $\endgroup$ – Clement C. Feb 9 '16 at 21:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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