Divergent Real Sequence to Positive Infinity/Examples/n^alpha

Example of Divergent Real Sequence to Positive Infinity
Let $\alpha \in \Q_{>0}$ be a strictly positive rational number.

Let $\sequence {a_n}_{n \mathop \ge 1}$ be the real sequence defined as:


 * $a_n = n^\alpha$

Then $\sequence {a_n}$ is divergent to $+\infty$.

Proof
We are to demonstrate that $n^\alpha \to +\infty$ as $n \to \infty$.

Let $H \in \R_{>0}$ be given.

We need to find $N \in \N$ such that:


 * $\forall n > N: n^\alpha > H$

That is:


 * $\forall n > N: n > H^{1 / \alpha}$

We choose $N = H^{1 / \alpha}$.

The result follows.