Editing Subadditivity (section)

==Definitions==
A subadditive function is a [[function (mathematics)|function]] <math>f \colon A \to B</math>, having a [[Domain of a function|domain]] ''A'' and an [[partial order|ordered]] [[codomain]] ''B'' that are both [[closure (mathematics)|closed]] under addition, with the following property:
<math display="block">\forall x, y \in A, f(x+y)\leq f(x)+f(y).</math>

An example is the [[square root]] function, having the [[non-negative]] [[real number]]s as domain and codomain:
since <math>\forall x, y \geq 0</math> we have:
<math display="block">\sqrt{x+y}\leq \sqrt{x}+\sqrt{y}.</math>

A [[sequence]] <math>\left \{ a_n \right \}_{n \geq 1}</math> is called '''subadditive''' if it satisfies the [[inequality (mathematics)|inequality]]
<math display="block"> a_{n+m}\leq a_n+a_m</math>
for all ''m'' and ''n''. This is a special case of subadditive function, if a sequence is interpreted as a function on the set of natural numbers.

Note that while a concave sequence is subadditive, the converse is false. For example, arbitrarily assign <math>a_1, a_2, ...</math> with values in <math>[0.5, 1]</math>; then the sequence is subadditive but not concave.