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#### issacnewton

##### Member

- Jan 30, 2012

- 61

Let \( X=(x_n) \) be a sequence of positive real numbers such that \( \mbox{lim }(x_{n+1}/x_n) = L > 1 \). Show that \( X \) is not bounded sequence

and hence is not convergent.

I am using negation of the goal. So I assumed that the sequence is bounded. In the limit definition, by using \( \varepsilon =L-1 \) I could show that

\[ \exists n_1 \in \mathbb{N}\;\forall \;n\geqslant n_1 \; (x_n < x_{n+1}) \]

Can people give some more hints ?