Absolute Value of Pearson Correlation Coefficient is Less Than or Equal to 1

Theorem
Let $X$ and $Y$ be random variables.

Let the variances of $X$ and $Y$ exist and be finite.

Then:


 * $\size {\map \rho {X, Y} } \le 1$

where $\map \rho {X, Y}$ denotes the Pearson correlation coefficient of $X$ and $Y$.

Proof
So:


 * $\size {\map \rho {X, Y} } \le 1$