- #1
issacnewton
- 1,026
- 36
here is the problem from Bartle's book
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 ?
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 ?