Beta Function is Defined for Positive Reals

Theorem
Let $x, y \in \R$ be real numbers.

Let $\map \Beta {x, y}$ be the Beta function:
 * $\ds \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$

Then $\map \Beta {x, y}$ exists provided that $x, y > 0$.

Proof
Consider the following inequalities, valid for $0 < t < 1$:

Then:

and similarly:

The result follows from the Comparison Test for Improper Integral.