Sequence of Powers of Reciprocals is Null Sequence

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

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

Then $$\left \langle {x_n} \right \rangle$$ converges to zero.

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

Then $$\left \langle {x_n} \right \rangle$$ converges to zero.

Proof
Let $$\epsilon > 0$$.

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

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