Stalk of inverse image sheaf

Question

I am trying to learn algebraic geometry from Hartshorne's book. I have a question regarding the stalk of the restricted sheaf. Let $X$ be a topological space and $Z$ a subspace of $X$. Consider the inclusion map $Z \hookrightarrow X$. Let $\mathcal{F}$ be a sheaf on $X$. Let $P$ be a point in $Z$. At the end of the first section in the second chapter, it is mentioned that the stalk of the restricted sheaf $\mathcal{F} \mid_{Z}$ at $P$ is equal (isomorphic) to the stalk of $\mathcal{F}$ at $P$. I tried to prove this statement from the definition of stalk and eventually ended up with direct limit of direct limits. Is there a better way to see this statement using the results of the first section?