Are there some strategies to prove a set has measure zero?

I'm still confused with subsets of $\mathbb{R}^n$ with measure zero. I mean, I know the definition very well: a subset $A$ of $\mathbb{R}^n$ has measure zero if for every $\epsilon > 0$ given there's an enumerable cover $\{U_i : i \in \Bbb N\}$ of $A$ by closed or open rectangles such that $\sum_{i=1}^\infty \operatorname{vol}(U_i)<\epsilon$.

That's fine, but finding this collection depending on $\epsilon$ seem to be tricky. The sequence of the partial sums $\sum_{i=1}^k\operatorname{vol}(U_i)$ should converge, so that this already restricts how should we pick the collection. And besides that, in general I still couldn't get how to do it.

When proving continuity, limits, integrability I have a strategy. For continuity (in $\Bbb R^n$) of a function $f$ for example at a point $a$, I write $|f(x)-f(a)|$ and try to bound this in terms of $|x-a|$ and things that I know I can bound without refering to $x$. For integrability, I do the same for the difference $U(f,P)-L(f,P)$. Well, these are just examples of cases I have an idea of a path to start, obviously each case requires a different thought, but I have an starting point.

What about proving a set has measure zero? What should be a good starting point? Is there some strategy we can use in these cases? I think there aren't any strategies, but I think it's valid asking.

Thanks very much in advance!

Two common strategies to prove that a set $A$ has measure $0$ are:

1. Find a set $B$, $A\subseteq B$, which also has measure $0$, but is easier to work with. (This is common in probability theory -- it amounts to showing that some necessary condition for event $A$ is satisfied with probability $0$, which of course implies that $A$ itself occurs with probability $0$.)

2. Find a decreasing sequence $(A_n)_{n\in\mathbb{N}}$ such that $A\subseteq A_n$ for all $n$ and $\mu(A_n)\rightarrow0$. (Of course, this is really (1) in disguise, as then $A\subseteq\bigcap_n A_n$, and $\mu(\bigcap_n A_n)\leq \mu(A_k)$ for all $k$.)

I realize that these aren't very concrete... but, every problem of this nature really has its own complications, so I don't think I can be more specific!