# Real analysis: why usually first limits of sequences and then limits of functions?

I notice that all of the analysis books that I've studied start from dealing with limits of sequences and only then move on to limits of functions.

Does this kind of approach have any particular advantage from an educational point of view? Or maybe it follows the historical development of the subject?

• I should also say that my professor made a point that she would do limits of functions first. – Dal Nov 11 '14 at 16:45
• By "successions" do you mean "sequences"? – Santiago Canez Nov 11 '14 at 17:51
• @SantiagoCanez Yes. – Dal Nov 11 '14 at 22:04
• To be clear, when we say "limits of functions", do you mean something like $\lim_{x \to 0} \frac{\sin x}{x}$, or do you mean something like $\lim_{n \to \infty} (1+x/n)^n$? I think you mean the former (that the variable under the limit is a continuous variable, and the thing to the right of the limit is a real valued function of that variable) but @Tom Au seems to me to be answering the latter (the thing to the right of the limit is a function of some other variable.) – David E Speyer Nov 12 '14 at 15:55
• One important reason is that the "limit of function" you refer to is an example of a limit of a net. Limits of nets naturally generalize limits of sequences, and many proofs that work for sequences immediately provide proofs in net language by replacing "sequence" by "net". This is not as natural from the other point of view. – Jon Bannon Nov 12 '14 at 17:05

## 2 Answers

One reason for this is that many beginning analysis books start with a rigorous development of the real numbers. If the chosen foundation is based on Cauchy sequences of rational numbers, then limits of sequences are logically prior to limits of functions of real variables.

• I think you've got a point. – Dal Nov 12 '14 at 22:19

"Sequences" refer to sequences of numbers.

"Numbers" are easier to deal with than functions, hence limits of sequences, versus limits of functions. (For most people, anyway.)

• You are understanding the question differently than I am; see my comment under the main post. – David E Speyer Nov 12 '14 at 15:58