Reciprocal of Null Sequence
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Theorem
Let $\sequence {x_n}$ be a sequence in $\R$.
Let $\forall n \in \N: x_n > 0$.
Then:
- $x_n \to 0$ as $n \to \infty$ if and only if $\size {\dfrac 1 {x_n} } \to \infty$ as $n \to \infty$
Corollary
- $x_n \to \infty$ as $n \to \infty$ if and only if $\size {\dfrac 1 {x_n} } \to 0$ as $n \to \infty$
Proof
Suppose $x_n \to 0$ as $n \to \infty$.
Let $H > 0$.
So $H^{-1} > 0$.
Since $x_n \to 0$ as $n \to \infty$:
- $\exists N: \forall n > N: \size {x_n} < H^{-1}$
That is:
- $\size {\dfrac 1 {x_n} } > H$.
So:
- $\exists N: \forall n > N: \size {\dfrac 1 {x_n} } > H$
and thus:
- $\sequence {\size {\dfrac 1 {x_n} } }$ diverges to $+\infty$.
$\Box$
Suppose $\size {\dfrac 1 {x_n} } \to \infty$ as $n \to \infty$.
By reversing the argument above, we see that $x_n \to 0$ as $n \to \infty$.
$\blacksquare$
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
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: Exercise $\S 4.29 \ (4)$