Eutrophication die, the bacterial degradation of their biomass consumes the oxygen in the water, thereby creating the state of hypoxia. According to Ullmann's Encyclopedia, "the primary limiting factor for eutrophication is phosphate." The availability of phosphorus generally promotes excessive plant growth and decay, favouring simple algae and plankton over other more complicated plants, and causes a severe reduction in water quality. Phosphorus is a necessary nutrient for plants to live, and is the limiting factor for plant growth in many freshwater ecosystems. Phosphate adheres tightly to soil, so it is mainly transported by erosion. Once translocated to lakes, the extraction of
Proofs involving the addition of natural numbers induction on "a", and so we have proved the base case "b" = 1. Now, suppose that for all natural numbers "a", we have "a" + "b" = "b" + "a". We must show that for all natural numbers "a", we have "a" + "S"("b") = "S"("b") + "a". We have This completes the induction on "b". Proofs involving the addition of natural numbers This article contains mathematical proofs for some properties of addition of the natural numbers: the additive identity, commutativity, and associativity. These proofs are used in the article Addition of natural numbers. This article will use the