I will be teaching an introductory analysis course in the coming semester. In it the students will learn about limits of real sequences, and then will learn about limits of functions in terms of sequences.
More precisely, we will say that $\lim_{x\to a+}f(x) = L$ if whenever $(x_n)$ converges to $a$ with $x_n>a$ for all $n$, we have $\lim_{n\to \infty} f(x_n) = L$. Likewise, we will say that $\lim_{x\to a-}f(x) = L$ if whenever $(x_n)$ converges to $a$ with $x_n< a$ for all $n$, we have $\lim_{n\to \infty} f(x_n) = L$. Then we say that $\lim_{x\to a} f(x) = L$ if both $\lim_{x\to a+}f(x) = L$ and $\lim_{x\to a-}f(x) = L$.
The students will have already seen the $\varepsilon$-$\delta$ definitions of limits of functions in their calculus course. The question then is, how to properly motivate this second (equivalent) definition of limits of functions?
Are there any arguments which become significantly easier when using the sequence definition of limits of functions in place of the $\varepsilon$-$\delta$ definition? (These should be elementary enough to be understood by first year Mathematics undergraduates.)
For instance, I suppose that once one has the Algebra of Limits for sequences, one gets the Algebra of Limits for functions for free. But I'm not convinced that much is to be gained from doing things this way around.
Edit: Thanks for the answers and comments so far. It seems many people are in favour of teaching the sequence definition of limits alongside the $\varepsilon$-$\delta$ definition. I agree that it should be useful to be aware of both definitions. To be certain of this, however, I would still like to see an example of a proof which is simpler when using the sequence definition.
Best Answer
Interestingly enough, as far as I remember, we did limits of sequences first (without functions) and then limits of functions (with $\varepsilon-\delta$) and then, as a remark, the connection mentioned above behind the very same Iron Curtain. Actually, in the end of our "limits course", our professor either did limits over nets, or stopped one step short of it: he certainly said all necessary words and made it clear that to talk about of a limit of a mapping, you just need some set of "catchers" in the range space and some set of "tails" in the argument space. That was a bit tough in the beginning but paid off nicely when doing Riemann integration where the tails are either partitions of small mesh or partitions subordinated to a fixed partition. I still find this abstract view rather enlightening; much more enlightening that the lemma in question, which, IMHO, only makes the concept more confusing (though is quite useful as a technical tool).
The main reason for this opinion is that this abstract view is unifying: all notions of limit that the students will ever meet fall under this idea, only the choices of catchers and tails vary and only one magic phrase is ever needed: "For every catcher, there is a tail whose image is contained in the catcher". The lemma you mentioned is separating: if used as a definition rather than a remark, it creates an impression that there are many ad hoc concepts of limits that all have to be understood and memorized separately, creating quite a mess in the student's head.
The $\varepsilon-\delta$ definition is already hard because it mixes the limit concept and the technical descriptions of the catchers and the tails, i.e., 3 things that can be easily separated and on which you can train the students one by one if you start with the abstract view. To be honest, I haven't tried it myself in the USA yet but I certainly will when teaching freshman analysis (so far it was either business calculus, where the game is never worth the candles, or advanced courses where the concept of limit was assumed to be well-known already).