theorem, composition of monotonically strictly increasing function with utility function

Theorem

Let $O$ be a set of outcomes, and let $\succsim$ be a complete, reflexive, and transitive preference relation over $O$. Suppose $u$ is a utility function representing $\succsim$. Then for every monotonically strictly increasing function $v : \mathbb{R} \to \mathbb{R}$, the composition $v \circ u$ defined by

is also a utility function representing $\succsim$.

Notes

The utility function is often called an ordinal function as a result of this theorem, as it represents only the order of preferences between outcomes. Numerical values a utility function associates with outcomes have no significance, and do not represent in any way the "intensity" of the player's preferences.

References

1. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, p. 11.