Limit Superior/Examples/Farey Sequence

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Example of Limit Superior

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 superior of $\sequence {a_n}$ is given by:

$\displaystyle \map {\limsup_{n \mathop \to \infty} } {a_n} = 1$


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 superior:

$\displaystyle \map {\limsup_{n \mathop \to \infty} } {a_n} = 1$

$\blacksquare$


Sources