Real Null Sequence/Examples/n^alpha x^n/Lemma

Lemma for Real Null Sequence: $n^\alpha x^n$
Let $\alpha \in \Q$ be a (strictly) positive rational number.

Let $x \in \R$ be a real number such that $\size x < 1$. There exists $N \in \N$ such that:


 * $\paren {1 + \dfrac 1 N}^{\alpha + 1} \, \size x \le 1$

Proof

 * $\forall n \in \N: \paren {1 + \dfrac 1 n}^{\alpha + 1} \, \size x > 1$
 * $\forall n \in \N: \paren {1 + \dfrac 1 n}^{\alpha + 1} \, \size x > 1$

Then:

But this contradicts Sequence of Powers of Reciprocals is Null Sequence.

Hence by Proof by Contradiction:


 * $\exists N \in \N: \paren {1 + \dfrac 1 N}^{\alpha + 1} \, \size x \le 1$