Show that no non-trivial open set in $R^n$ can have measure zero in $R^n$.
Attempt at the solution: I am having a lot of difficulty attempting this question, I have read a lot of material on measure zero and almost all of the questions posted on this forum regarding the topic, but I still can't seem to understand how to attempt this problem. I have written a proof that somewhat makes intuitive set to me, but I am pretty terrible at proof writing and trying to improve so please don't be sparing in your recommendations and criticism.
Let A be open set in $R^n$ of the form (a,b) s.t a $\not=$ b, we can choose intervals I for any $x_i$ $\in$ A :[$x_i - \epsilon $, $x_i + \epsilon$], we can make these intervals arbitrarily small, the problem occurs at the points a and b, we will need an interval that covers a and closest point to a that is included in the set, the smallest interval of this form would be [a-$\epsilon$, $x_1 + \epsilon$], this interval does not have measure zero by the definition of open sets ( since if a is arbitrarily close to $x_1$ we would arrive at a contradiction i.e A will no longer be open), which gives us that no non trivial open set can have measure zero. Thanks in advance