>Right now, the atoms in my fingers are made of electrons and nuclei.
>
>The force that holds them to each other is Electromagnetism.
>
>Light is also Electromagnetism

An electromagnetic wave, to be precise.

>Do you really think the Universe would also ensure that the Electromagnetic force not only causes interactions between local atoms, but between atoms trillions of light years apart?
>
>It seems like the EM force has a job description, one that's pretty essential for life (http://en.wikipedia.org/wiki/Anthropic_principle).
>
>Why it would have to have an infinite range?
>
>The forces that hold nuclei together don't have a range past the nucleus.
>
>Why doesn't the force that holds electrons in the atom have a similar range?

That's what scientists are investigating. It obviously does go over large distances, but the reason is not clear yet.

Just forget about the "infinite" - I think this was what started to get you thinking. The whole idea of saying that light has an "infinite range", or that gravitation has an "infinite range", is that it goes further than the typical distance between nucleons, or even atoms or molecules, and that it is known to go a great distance. Of course it won't go an "infinite distance" if space is finite. Well, actually it may, if space continues expanding indefinitely. But this is simply a way of saying that it is "unbounded" - no known limit.