So far, for my proof I have:
Let $(x_n)$ be a sequence of positive real numbers with the the property that $x_n+1/x_n < C\forall n\in\Bbb N$, where $0 < C < 1$.
(a) I know that I can prove this by induction. Since for the basis step I would get $x_n\leqslant C\forall n\in\Bbb N$. The inductive step is a bit tricky for me.
(b) Use the $e- K$ definition, and the result that $(x_n)$ converges to $0$. b) since I am stuck in part a I’m not sure what result to use to prove that the sequence converges to $0$, but I do know that I can take the $\ln$ of both sides when solving and then this may be useful for part a which would let me say in part (b) something about $x<1$ then $\ln(x)<0$.