I am facing difficulty in employing Cauchy's general principle of convergence to prove that a sequence is convergent.
Let's take this example:
Let $x_1 = 2$, and $x_{n+1} = \sqrt{x_n+20}$, for $n=1,2,3,\dots$
Prove that the sequence {$x_n$} converges.
I solved this using monotone convergence theorem. Using induction, it can be proved that the sequence is increasing and is bounded above by $5$. Thus, it is convergent.
But how do I prove this using Cauchy's principle, which states that, for each $\epsilon > 0$ there exists a positive integer $m$ such that
$|S_{n+p}-S_n|<\epsilon$, $\forall$ $n\geq m $ and $p\geq1$, $\{S_n\}$ being the sequence.