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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?

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  • $\begingroup$ I should also say that my professor made a point that she would do limits of functions first. $\endgroup$
    – Dal
    Nov 11, 2014 at 16:45
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    $\begingroup$ By "successions" do you mean "sequences"? $\endgroup$
    – Santiago Canez
    Nov 11, 2014 at 17:51
  • $\begingroup$ @SantiagoCanez Yes. $\endgroup$
    – Dal
    Nov 11, 2014 at 22:04
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    $\begingroup$ 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.) $\endgroup$
    – David E Speyer
    Nov 12, 2014 at 15:55
  • $\begingroup$ 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. $\endgroup$
    – Jon Bannon
    Nov 12, 2014 at 17:05

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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.

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  • $\begingroup$ I think you've got a point. $\endgroup$
    – Dal
    Nov 12, 2014 at 22:19
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"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.)

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    $\begingroup$ You are understanding the question differently than I am; see my comment under the main post. $\endgroup$
    – David E Speyer
    Nov 12, 2014 at 15:58

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