Reciprocal of Null Sequence/Corollary

Theorem

Let $\sequence {x_n}$ be a sequence in $\R$.

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

Then:

$x_n \to \infty$ as $n \to \infty$ if and only if $\size {\dfrac 1 {x_n} } \to 0$ as $n \to \infty$

Proof

Let $\sequence {y_n}$ be the sequence in $\R$ defined as:

$y_n = \size {\dfrac 1 {x_n} }$

From Reciprocal of Null Sequence:

$y_n \to 0$ as $n \to \infty$ if and only if $\size {\dfrac 1 {y_n} } \to \infty$ as $n \to \infty$

That is:

$\size {\dfrac 1 {x_n} } \to 0$ as $n \to \infty$ if and only if $x_n \to \infty$ as $n \to \infty$

$\blacksquare$

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: Exercise $\S 4.29 \ (4)$