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Convergent sequence

Let \{x_n\}_{n \in \mathbb{N}} be a sequence of positive real numbers such that

(1)   \begin{equation*} x_{n+m} \leq x_n + x_m \quad , \quad  m, n \in \mathbb{N} \end{equation*}

Prove that \left\{\dfrac{x_n}{n} \right\}_{n \in \mathbb{N}} converges.

Solution

Fix m and let n \geq m. Then, there exist k, r such that n=km+r where 0\leq r <m. Thus,

    \[\frac {x_n}{n} = \frac {x_ {km+r}}{n}\leq \frac {kx_ {m}}{n} + \frac {x_{r}}{n}\]

Letting n \rightarrow +\infty it follows that

    \[\limsup \frac {x_n}{n} \leq \frac {x_ {m}}{m} + 0\]

Since this holds forall m it follows that \displaystyle \limsup \frac {x_n}{n} \leq \liminf \frac {x_ {m}}{m} and the result follows.

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