Reciprocal of Null Sequence

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Theorem

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

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


Then:

$(1): \quad z_n \to 0$ as $n \to \infty$ iff $\left|{\dfrac 1 {z_n}}\right| \to \infty$ as $n \to \infty$
$(2): \quad z_n \to \infty$ as $n \to \infty$ iff $\left|{\dfrac 1 {z_n}}\right| \to 0$ 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, $\left|{\dfrac 1 {z_n}}\right| > H$.

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


  • Suppose $\left|{\dfrac 1 {z_n}}\right| \to \infty$ as $n \to \infty$.

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

$\blacksquare$


The statement:

$z_n \to \infty$ as $n \to \infty$ iff $\left|{\dfrac 1 {z_n}}\right| \to 0$ as $n \to \infty$

is proved similarly.

$\blacksquare$


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

$\blacksquare$


Also known as

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.


Sources