Help proving exercise on sequences in Bartle's Elements

real-analysis sequences-and-series self-learning

You have successfully unpacked the definition of the limit into the form that will become useful. You have already shown that there exists $m\in\mathbb N$ such that for all $n\geq m$ you have $$\frac{x_{n+1}}{x_n}<l+\epsilon=:r<1. \tag1$$

All you need now is to put some recursion to work. This is the core of the criterion: if the limit of ratios of successive elements is smaller than one, this means that each element of the sequence is smaller than the previous one by some fixed percentage. If you go multiple steps out, this percentage will be compounded, which is what gets you the bound.

To make that precise, note that multiple applications of the bound (1) will give you

  • $x_{m+1}<x_m\,r,$
  • $x_{m+2}<x_{m+1}\,r<x_m\,r^2,$
  • $x_{m+3}<x_{m+2}\,r<x_m\,r^3,$

and so on, so that a trivial induction step will give you $$x_{m+k}<x_m\,r^k.$$ This is essentially the bound you were after, if you set $C=x_m/r^m$.

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