# A Pair of Interesting Inequalities

In research with product partitions models, especially the Dirichlet process mixture, I've found a pair of inequalities useful. They are reproduced as theorems below. Does anyone know if these are special cases of a well-known theorem?

Theorem: Let $n_1$ and $n_2$ be positive integers such that $n_1 \geq n_2$. Then,

.

Proof: $n_1 \geq n_2 \geq 1$ implies

Theorem: Let $n_1$ and $n_2$ be positive integers such that $n_1 \geq n_2$. Then,

.

Proof: $n_1 \geq n_2 \geq 1$ implies