How do I show that a monomorphism $F \rightarrow G$ of sheaves induces an injection on stalks?
When showing that monomorphism is an injection on sets one uses the maps $x \mapsto a$ and $x \mapsto b$ where $a,b$ are some fix elements such that they have the same image under the monomorphism. Then by using the monomorphism property we must have $a = b$. But since we are dealing with sheaves we are supposed to construct a skyscraper sheaf and two maps that map "almost all" section of the skyscraper sheaf to $a$ and $b$ (respectively), or at least elements that are "connected" to $a$ and $b$ (since these are representations of elements in the stalk there are other options depending on which open set we are taking). When I say "almost all" and "connected" it is just my visualization of how it should be.
I'm not able to neither define the correct skyscraper sheaf nor the maps to $F$ (especially such that the maps commute with the restriction maps). I was hoping for this being true which might have helped me but now I am stuck and need some help with this problem.
Edit: I am still interested in how the problem will be solved by using a skyscraper sheaf.