Show $\mathbb {P}(\limsup\limits_{n\to\infty}Z_n/\ln n=1)=1$ for $Z_n$ iid $Exp(1)$

Question

$\mathbb {P}(\limsup\limits_{n\to\infty}Z_n/\ln n=1)=1$ for $Z_n$ iid $Exp(1)$

I guess I have to use Borel-Cantelli but I am struggling with especially one part. Shouldn't we have

$\sum_{n=0}^\infty \{Z_n/\ln n=1\} = 0 < \infty$ as $Z_n$ are continues Random Variables?