I am aware this is a very simple question.
Let $(M,u_d)$ be the inverse limit of the direct set of modules $(M_d,v_{ds})$ over a commutative ring $A$. If $u_d$ maps an element $x\in M$ to $0$ in $M_d$ for each $d$ how do I show that $x$ must be the $0$ element directly from the universal property? Do I have to look at the construction?
In other universal constructions, for example tensor products of modules $M$ and $N$ over a commutative ring $A$ with identity I can deduce that $x\otimes 0=0\otimes y=0$ in $M\otimes N$ easily from the fact that the map $M\times N\to M\otimes N$ is bilinear. But for inverse limit I haven’t figure out a way to determine if an element is $0$ if each image of the projection is $0$.