Superadditive

In mathematics, a sequence { a_n }, n ≥ 1, is called superadditive if it satisfies the inequality

for all m and n. The major reason for the use of superadditive sequences is the following lemma due to Michael Fekete.

Lemma: (Fekete) For every superadditive sequence { a_n }, n ≥ 1, the limit lim a_n/n exists and is equal to sup a_n/n. (The limit may be positive infinity, for instance, for the sequence a_n = log n!.)

Similarly, a function f is superadditive if

for all x and y in the domain of f.

For example, ${\displaystyle f(x)=x^{2}}$ is a superadditive function for nonnegative real numbers because the square of ${\displaystyle (x+y)}$ is always greater than or equal to the square of ${\displaystyle x}$ plus the square of ${\displaystyle y}$, for nonnegative real numbers ${\displaystyle x}$ and ${\displaystyle y}$.

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