Let $(a_{n})_{n \geq1}$ be a sequence of numbers such that $a_n\leq M$ for all $n \geq 1$ . Prove that

$$ \lim_{n\to\infty} \inf \{a_n,a_{n+1},...\} = \sup_{n \geq1} \inf\{a_n,a_{n+1},...\} $$

This is a homework problem.

But I'm really confused where to start. I'm an engineering student taking analysis course and have no previous background of rigorous proof. It'll be help full if someone can tell me how to approach the problem.

Thank you.

  • $\begingroup$ Is this course in real analysis one that all engineering students take, or did you choose it as an elective? I'm asking because usually students have taken other, more gentle proof-based math courses before taking real analysis to prepare them for the proofs they encounter in the course. If you are an engineering student, I doubt you've taken previous courses in pure math. $\endgroup$ – layman Feb 1 '15 at 13:25
  • $\begingroup$ @Vinith I think you should not accept such a quick answer. Unless, of course, you've already checked my tip step by step with pencil and paper. Dealing with 'sup' and 'inf' it is necessary familiarity and a good understanding the definition of supremum and the definition of infimum. This familiarity is achieved with practice various exercises. $\endgroup$ – MathOverview Feb 1 '15 at 15:05
  • $\begingroup$ @user46944 : No this is an elective course. I'm planning to work in optimization and machine learning, so I thought a good understanding of analysis would really be helpful. There are not many proof based math courses offered in my uni. I'm trying read the basics from "Introduction to analysis" by Arthur Mattuck. It would be of great help if you can advise me any good book to follow. $\endgroup$ – Vinith Feb 2 '15 at 14:29
  • $\begingroup$ @Elias : Using the hint given by you I was able to show both the limits coincide. I believe what I did was correct and hence I approved the answer.Thank you. $\endgroup$ – Vinith Feb 2 '15 at 15:48
  • $\begingroup$ @Vinith Understanding Analysis by Stephen Abbott is a good book. It's a gentle introduction to the subject, and it is what I used when I first studied Analysis. His book is conversational in tone (but still full of proofs and exercises). I liked it and you might, too. $\endgroup$ – layman Feb 2 '15 at 16:23

Consider the sequence of numbers $$ I_1=\inf\{a_1,a_2,\ldots, a_n,\ldots\}\leq a_1\leq M, \\ I_2=\inf\{a_2,a_3,\ldots, a_n,\ldots\}\leq a_2\leq M, \\ \vdots \\ I_n=\inf\{a_n,a_{n+1},a_{n+3}\ldots\}\leq a_n\leq M. $$ form non-empty sets $\{I_1,\ldots, I_n\}$, for each $n\in\mathbb{N}$, bounded by constant $M$. By definition, we have the supreme of the set is such that $$ \sup\{I_1,\ldots,I_n\}\geq I_n. $$ On the other hand, as $ I_1\leq I_{2}\leq \ldots \leq I_{n}\leq \ldots \leq M $ we have $$ \sup_{n\geq 1}\inf\{a_n,a_{n+1},a_{n+2},\ldots \}=\sup_{n\geq 1}\{I_1,\ldots, I_n\}=I_n\leq M. $$ So what can you say about $\lim_{n\to \infty}S_n$,for $ S_n=\sup\{I_n, I_{n+1},I_{n+2},\ldots \}, $ and $\lim_{n\to\infty}I_n$ ?


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