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.

I am studying analysis and having difficulties with $\inf$, $\sup$, $\liminf$ and $\limsup$ as you can see from the title. As discussed in a number of math.SE questions, definitions for sequences, functions and sets differ from each other. I sense, I have to tackle with some problems which are really useful for the pedagogic purposes. Also, I want to learn definitions in all senses.

So, can anyone suggest a reference which have a number of good exercises?


share|improve this question
add comment

2 Answers 2

up vote 4 down vote accepted

Is there a particular text you are currently using? That might help in terms of suggesting supplementary texts (rather than possibly a duplicate!)

Here are some lecture notes on $\sup$ and $\inf$, and here are some lecture notes on $\lim \sup$ and $\lim\inf$, both in pdf, with some exercises included, which may serve as handy references.

Also see the site "Interactive Real Analysis" which includes both tutorials and exercises, for many topics in Analysis, including $\lim\inf, \lim\sup, \sup, \inf$, and more! This you can access immediately, or from anywhere you can access the internet.

I'd also explore this site for particular problems posted, and their solutions, which will give you some practice with and approaches to dealing with questions of this sort.

For a textbook reference: Serge Lang's Undergraduate Analysis covers these concepts in detail, and there's an accompanying problem and solution text for even more exercises, for practice.

share|improve this answer
I am an engineering graduate who is taking real analysis class and textbook is Real Analysis of Folland. Although I gained some familiarity to use these tools in proofs, I sense still I need some particular exercises to gain more intuition not only for $\liminf$ and $\limsup$ but also for $\sup$ and $\inf$ (because still I am surprising sometimes about their uses in proofs). Thank you very much! –  Deniz Jan 7 '13 at 1:19
John. Yes, I understand. Some texts go through the subject matter rather quickly (hand-waving, so to speak), and the concepts are not all that intuitive to many students. I commend your efforts to tackle what you're struggling with! –  amWhy Jan 7 '13 at 1:24
thanks for references and encouragement! :) –  Deniz Jan 7 '13 at 1:29
@amWhy: Nice references too! +1 –  Amzoti May 8 '13 at 2:34
add comment

I may suggest the book of Karl R. Stromberg "An Introduction to Classical Real Analysis".

share|improve this answer
thanks for the reference! –  Deniz Jan 7 '13 at 0:03
you're welcome! –  беркай Jan 7 '13 at 0:05
add comment

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.