Sequence of Powers of Reciprocals is Null Sequence

Theorem
Let $r \in \Q_{>0}$ be a strictly positive rational number.

Let $\sequence {x_n}$ be the sequence in $\R$ defined as:
 * $x_n = \dfrac 1 {n^r}$

Then $\sequence {x_n}$ is a null sequence.

Real Index
If $r \in \R_{>0}$ is a strictly positive real number, the same result applies.

However, the result is specifically stated for a rational index, as this definition is used in the course of derivation of the existence of a power to a real index.

Proof
Let $\epsilon \in \R_{>0}$.

We need to show that:
 * $\exists N \in \N: n > N \implies \size {\dfrac 1 {n^r} } < \epsilon$

That is, that $n^r > 1 / \epsilon$.

Let us choose $N = \ceiling {\paren {1 / \epsilon}^{1/r} }$.

By Reciprocal of Strictly Positive Real Number is Strictly Positive and power of positive real number is positive, it follows that:
 * $\paren {\dfrac 1 \epsilon}^{1/r} \gt 0$

Then by Positive Power Function on Non-negative Reals is Strictly Increasing:
 * $\forall n > N: n^r > N^r \ge 1 / \epsilon$

Also see
This result and Sequence of Powers of Number less than One‎ are sometimes referred to as the basic null sequences.


 * Limit at Infinity of $x^n$