## Barbalat’s Lemma

Prove that if we have a function which is uniformly continuous on with then .

**Proof:** Suppose there exists such that . Moreover, we can suppose increasing and that the difference is large enough for each . Take . Then there exists such that we have . From uniform continuity, there exists such that .

It easily follows that if .

Therefore . This contradicts .

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Categories: Analysis
continuous, uniformly

Hi Beni Bogosel,

I needed a quick proof of Barbalat’s Lemma to put in my Master thesis, so I reproduced the one you exposed here in this post– of course, with the appropriated references to your blog. Hope you don’t mind 😉

(Please mail me if this represents any problem for you)

I’m glad you found the proof useful. It is indeed a proof I discovered myself, but I cannot take credit for it, because the ideas I used are quite standard, and I think that most people who would try and prove Barbalat’s Lemma would go through the same steps I did. Thank you for reading my blog. 🙂

I think you should specify that the sequence is increasing or tends to infinity.

Yes, the sequence goes to and can be chosen to be increasing.