How do reversible reactions reach an equilibrium?

Reversible reactions are reactions where the products are able to react themselves into the reactants. At the beginning of the reaction, there will be more reactants than products. But as the reaction goes on the amount of reactants will be decreasing and the amount of products will be increasing. So after a period of time, the amount of products and reactants will be the same and stay at this level. Because the products are able to react back into the reactants, there will be no change in the amount of products and reactants. The rates of reaction for the forward and backward reaction are the same which means that the reaction has reached an equilibrium.