Show that a subset $E$ of $R$ has measure zero iff there exists a sequence of intervals $\{I_{n}\}$ such that $a)\sum\ m(I_{n})<\infty; $ and $b)$ $x $ in $E$ implies $x$ lies in infinitely many $I_{n}$'s .
I am sorry for formatting, I am newcomer to this forum, most of the things seems pretty new to me.
I am comfortable proving one direction from right to left, where I use the Borel-Cantelli lemma to get the result.
While attempting left to right, we can easily get the covering of the set $E$ which has measure less than any chosen epsilon ( by def of outer measure). My problem is showing for each $x$ in $E$ lies in infinitely many of $I_{n}$'s. That is essentially proving $x$ belong to $\limsup I_{n}$ (I think). I can imagine $x$ lying in many of $I_{n}$ but I do not know how can I be rigorous on that point.