Show that the set of units, U, of R forms a group under multiplication.

My thoughts... U is a group since if we multiply a unit by a unit we get a unit. (not sure if true and would probably need a proof...)

Formally, let a,b be units. Then a|1 and b|1. So ab|1, hence ab is a unit. So is a group under multiplication...

Just a draft of what i think the answer is. Pretty sure there's more to it though. Am i on the right track?