showing $mu(limsup_{ntoinfty} E_n) = 0.$

As Bernard comments, the first part is actually the statement of Borel-Cantelli lemma.

For the second part, as Stella mentions, we may want to find a series which is not convergent but the series consists of square of its terms is convergent. The series $sum_{n=1}^infty frac{1}{n}$ will satisfy this condition.

So we need to construct a sequence of sets ${E_n}$ s.t. $mu(E_n)=frac{1}{n}, forall nge1.$ Also, we want that $mu(limsup_{ntoinfty} E_n) neq 0.$

To construct a concrete example, we may need a proposition.

Proposition: For every real number x, there exists a sequence $x_1,x_2,x_3,...$ of integers such that $$x=x_1+frac{x_2}{2!}+frac{x_3}{3!}+cdotcdotcdot ,,$$ where $x_1$ can be any integer, but for $nge2,x_nin{0,1,...,n-1}.$ Furthermore, if we require that the partial sums be strictly smaller than x, then such a representation is unique.

A proof can be found here: https://blogs.ams.org/mathgradblog/2017/09/20/real-numbers-base-factorials-by-product/

From the proposition and its proof, it's easy to see that, if $xin(0,1)$, being an irrational number, then we can find an unique sequence $x_1,x_2,x_3,...$ s.t. $x_1=0$ and $$x=frac{x_2}{2!}+frac{x_3}{3!}+cdotcdotcdot ,,$$ with $x_nin{0,1,...,n-1},forall nge2.$

Also note that, if $x$ is an irrational number, then the corresponding series will consist of infinitely many terms. (N.B. This is crucial to our construction, which makes sure the construction process will not terminate in finite steps.)

Our example will be given in the language of probability space, but in essence, it can be easily converted into the language of general measure theory.

We define the event $E_1={text{pick a random irrational number $x$ in $(0,1)$, $x_1$ is $0$}. }$. Then it's clear that $P(E_1)=1.$

For each $nge 2,$ define event $E_n={text{pick a random irrational number $x$ in $(0,1)$, $x_n$ is $0$}. }$. Then it's also clear that $P(E_n)=frac{1}{n}.$

Note that these events are mutually independent.

By the second Borel-Cantelli lemma(c.f. https://en.wikipedia.org/wiki/Borel–Cantelli_lemma#Converse_result), we conclude that $P(limsup_{ntoinfty} E_n)=1neq 0.$

So, even though $sum_{n=1}^infty mu(E_n)^2 < infty$, we may still have $mu(limsup_{ntoinfty} E_n)=1neq 0.$

$$tag*{$blacksquare$}$$