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 $n \to \infty$ $\size {\dfrac 1 {x_n} } \to 0$ 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 $n \to \infty$ $\size {\dfrac 1 {y_n} } \to \infty$ as $n \to \infty$

That is:


 * $\size {\dfrac 1 {x_n} } \to 0$ as $n \to \infty$ $x_n \to \infty$ as $n \to \infty$