Given $Z_n\rightarrow ^p b$, and if $g$ is continuous at $b$, then $g(Z_n)\rightarrow^p g(b)$.
That's the context of the problem, but my real issue is this. The definition of continuity is $\forall\epsilon\exists\delta:|Z_n-b|<\delta\Rightarrow |g(Z_n)-g(b)|<\epsilon$. This implies that $|Z_n-b|<\delta$ is a subset of $|g(Z_n)-g(b)|<\epsilon$. I'm struggling to understand why? If there are only some deltas such that the implication holds, isn't the implication a subset of the left side? Like say it holds for deltas below 5 and not for deltas more than 5. Then surely $|g(Z_n)-g(b)|<\epsilon \subseteq|Z_n-b|<\delta$? What's wrong with this reasoning, if anything?