Beta Function is Defined for Positive Reals

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

Let $\Beta \left({x, y}\right)$ be the Beta function:
 * $\displaystyle \Beta \left({x, y}\right) := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \left({1 - t}\right)^{y - 1} \ \mathrm d t$

Then $\Beta \left({x, y}\right)$ 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.