Sequence of Powers of Reciprocals is Null Sequence

Theorem
Let $$r \in \Q^*_+$$ be a strictly positive rational number.

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

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

Real Index
If $$r \in \R^*_+$$ 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 $$x_n = \frac 1 n$$.

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

Proof
Let $$\epsilon > 0$$.

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

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 $$1 \in \Q^*_+$$ the result follows.