Let $\lambda$ counting measure on $\mathbb{N}$, and let
$\mu (E) = \sum_{n\in E}2^{-n}$
defined for $E \subset\mathbb{N}$. How to prove that $\lambda$ is not absolutely continuous with respect to $\mu$? I find it strange because in this case I should find a set $E \subset\mathbb{N}$ in which $\mu (E) = 0$ and $\lambda (E) \neq 0$. I'll leave the exercise photo below because I may have misunderstood
You have mis-interpreted the statement. $\lambda <<\mu$ for sure but it is not true that $\lambda (E) \to 0$ as $\mu (E) \to 0$. For example, $\mu (\{n\})=\frac 1 {2^{n}} \to 0$ but $\lambda (\{n\})=1$ for all $n$. [This there is no $\delta$ corresponding to $\epsilon =1$].