Dougall's Hypergeometric Theorem/Corollary 3/Lemma

Lemma for Dougall's Hypergeometric Theorem/Corollary 3

 * $\ds \lim_{z \mathop \to \infty} \dfrac {\paren {x + z + n + 1}^{\overline y} } {\paren {z+ n + 1}^{\overline y} } = 1$

where $x^{\overline y}$ denotes the $y$th rising factorial of $x$.

Proof
From Properties of Limit at Infinity of Real Function: Product Rule, we have:

From L'Hôpital's Rule:Corollary 2, we have:
 * $\ds \lim_{z \mathop \to a^+} \frac {\map f z} {\map g z} = \lim_{z \mathop \to a^+} \frac {\map {f'} z} {\map {g'} z}$

In the present example, for the kth limit, we have $\ds \map {f_k} z = \paren {x + z + n + y}$ and $\ds \map {g_k} z = \paren {z + n + y}$

Therefore taking the derivative of the numerator $\map {f_k} z$ and denominator $\map {g_k} z$ $z$, we obtain:
 * $\ds \lim_{z \mathop \to \infty}\paren {\dfrac {\paren {x + z + n + y} } {\paren {z + n + y} } } = \lim_{z \mathop \to \infty} \dfrac 1 1 = 1$

Therefore: