Real Null Sequence/Examples/n^alpha over y^n

Example of Real Null Sequence
Let $\alpha \in \R$ be a (strictly) positive real number.

Let $y \in \R$ be a real number such that $\size y > 1$.

Let $\sequence {a_n}_{n \mathop \ge 1}$ be the real sequence defined as:
 * $\forall n \in \Z_{>0}: a_n = \dfrac {n^\alpha} {y^n}$

Then $\sequence {a_n}$ is a null sequence:


 * $\ds \lim_{n \mathop \to \infty} \dfrac {n^\alpha} {y^n} = 0$

Proof
Let $y = 1 + x$.

Then: