Reciprocal of Null Sequence

Theorem
Let $$\left \langle {z_n} \right \rangle$$ be a sequence in $\C$.

Let $$\forall n \in \N: z_n > 0$$.

Then:
 * $$z_n \to 0$$ as $$n \to \infty$$ iff $$\left|{\frac 1 {z_n}}\right| \to \infty$$ as $$n \to \infty$$;
 * $$z_n \to \infty$$ as $$n \to \infty$$ iff $$\left|{\frac 1 {z_n}}\right| \to 0$$ as $$n \to \infty$$.

Real Numbers
If $$\left \langle {x_n} \right \rangle$$ is a sequence in $\R$, the same applies.

Proof

 * Suppose $$z_n \to 0$$ as $$n \to \infty$$.

Let $$H > 0$$.

So $$H^{-1} > 0$$.

Since $$z_n \to 0$$ as $$n \to \infty$$, $$\exists N: \forall n > N: \left|{z_n}\right| < H^{-1}$$.

That is, $$\left|{\frac 1 {z_n}}\right| > H$$.

So $$\exists N: \forall n > N: \left|{\frac 1 {z_n}}\right| > H$$ and thus $$\left \langle {\left|{\frac 1 {z_n}}\right|} \right \rangle$$ diverges to infinity.


 * Suppose $$\left|{\frac 1 {z_n}}\right| \to \infty$$ as $$n \to \infty$$.

By reversing the argument above, we see that $$z_n \to 0$$ as $$n \to \infty$$.

The statement "$$z_n \to \infty$$ as $$n \to \infty$$ iff $$\left|{\frac 1 {z_n}}\right| \to 0$$ as $$n \to \infty$$" is proved similarly.


 * If $$\left \langle {x_n} \right \rangle$$ is a sequence in $\R$, the same argument can be used directly.

Note
Some sources call this the "reciprocal rule", but as that name is used throughout mathematical literature for several different concepts, its use is not recommended.