Since I started to learn a bit about set theory, I am paying more attention as where I am or not using the axiom of choice in analysis.
I had a small exercise about measure that asks me to show that on the real line, a countable number of points $C$ has measure 0. So since $C = \{c_n\}_{n\ge 1}$, for every $\epsilon$ we can cover $C$ by open intervals $I_n = (c_n-\frac{\epsilon}{2^n}, c_n+\frac{\epsilon}{2^n})$ such that and $\sum_{n\ge 1} \mid I_n \mid < C \epsilon$, for some constant C. But to define these $I_n$, do I use the axiom of choice? Dependent choice? Countable choice?
No, from the enumeration $c_n, n \in \Bbb N$ you produce explicit intervals. No choice is involved.