Limit Inferior/Examples/Farey Sequence
Jump to navigation
Jump to search
Example of Limit Inferior
Consider the Farey sequence:
- $\sequence {a_n} = \dfrac 1 2, \dfrac 1 3, \dfrac 2 3, \dfrac 1 4, \dfrac 2 4, \dfrac 3 4, \dfrac 1 5, \dfrac 2 5, \dfrac 3 5, \dfrac 4 5, \dfrac 1 6, \ldots$
The limit inferior of $\sequence {a_n}$ is given by:
- $\ds \map {\liminf_{n \mathop \to \infty} } {a_n} = 0$
Proof
Let $L$ be the set of all real numbers which are the limit of some subsequence of $\sequence {a_n}$.
From the definition of $F$ we have that:
- $\forall n \in \N_{>0}: 0 < a_n < 1$
From Lower and Upper Bounds for Sequences we have that:
- $L \subseteq \closedint 0 1$
Consider the subsequences:
- $(1): \quad \sequence {a_{n_r} } = \dfrac 1 2, \dfrac 1 3, \dfrac 1 4, \dfrac 1 5 \to 0$ as $n \to \infty$
- $(2): \quad \sequence {a_{n_s} } = \dfrac 1 2, \dfrac 2 3, \dfrac 3 4, \dfrac 4 5 \to 1$ as $n \to \infty$
Hence by definition of limit inferior:
- $\ds \map {\liminf_{n \mathop \to \infty} } {a_n} = 0$
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 5$: Subsequences: Lim sup and lim inf: $\S 5.14$: Example $\text {(iii)}$