To prove this can I proceed by contradiction and say, assume that $f: \mathbb{N} \to (0,1)$ is surjective, this would imply that $f(\mathbb{N})$ is equinumerous to $(0,1)$, but since $\mathbb{N}$ is countable, and $(0,1)$ is uncountable then they cannot be equinumerous and thus a contradiction, therefore $f: \mathbb{N} \to (0,1)$ is NOT surjective.
What are some other ways if this is correct?