I am teaching introductory real analysis this term and realize that my students have problem coming up with sequence in some arguments in real analysis. Let's take this example:
Theorem: Given a function $ f: [a,b] \to \mathbb R$ and $x_0\in [a,b]$. If for all sequence $\{a_n\}_{n=1}^\infty$ in $[a,b]\setminus \{x_0\}$ which converges to $x$, $\{f(a_n)\}_{n=1}^\infty$ converges to $L$. Then $f$ has a limit $L$ at $x$.
Proof Assume the contrary that $f$ does not have limit $L$ at $x_0$. Then there is $\epsilon_0 >0$ such that for all $\delta>0$, there is $x\in [a,b]\setminus\{ x_0\}$ so that $|x-x_0|<\delta$ and $|f(x) - L|\ge \epsilon_0$.
Then the next step is to choose (e.g.) $\delta = 1/n$ and come up with a sequence $\{x_n\}_{n=1}^\infty$ with $|x_n - x_0|<1/n$....
This step involves the (countable) Axiom of choice. Every time I perform a similar argument in class, they seem to understand it. But they are failing in the HW/midterm. It seems that their complaint is that they cannot see how to choose the sequence.
It seems to me that their confusion is legit, since this is the major reason why the Axiom of choice got some criticisms.
I would just throw "Hey! This is Axiom of Choice!" to them, but (1) this is not how we study real analysis here, where they don't have a solid background in set theory, and (2) that does not seem to help them understand the concept.
So my question is, how do we in general motivate the (implicit) use of AC in real analysis?