Sequence of Powers of Reciprocals is Null Sequence

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

Let $\left \langle {x_n} \right \rangle$ be the sequence in $\R$ defined as $x_n = \dfrac 1 {n^r}$.

Then $\left \langle {x_n} \right \rangle$ 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.

Corollary
Let $\left \langle {x_n} \right \rangle$ be the sequence in $\R$ defined as $x_n = \dfrac 1 n$.

Then $\left \langle {x_n} \right \rangle$ is a null sequence.

Proof
Let $\epsilon > 0$.

We need to show that $\exists N \in \N: n > N \implies \left|{\dfrac 1 {n^r}}\right| < \epsilon$.

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

Let us choose $N = \left({1/\epsilon}\right)^{1/r}$.

Then $\forall n > N: n^r > N^r = 1 / \epsilon$.

Proof of Corollary
$n = n^1$ from the definition of power and as $1 \in \Q_{>0}$ the result follows.

Also see

 * Limit at Infinity of x^n