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

Example of Real Null Sequence
Let $\alpha \in \R$ be a (strictly) positive rational 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 $x = \dfrac 1 y$.

Thus:
 * $a_n = n^\alpha x^n$

such that $x < 1$.

Then Real Null Sequence: $n^\alpha x^n$ applies.