Shortcut to recognize a cauchy sequence?

Question

I know kinda shortcut for uniformly continuous functions (which is Cauchy criterion) by seeing if the derivative of the function is bounded or not, so I was wondering if there is a shortcut or trick or tip for recognizing that the sequence is a Cauchy sequence without using the formal definition?

Answer 1

What you are referring to is called the Lipschitz Criteria.

If the function has a bounded derivative then By Lagrange's Mean value theorem

$f(x)-f(y)=f'(c)(x-y)\,\,,c\in(x,y)\implies |f(x)-f(y)|=|x-y||f'(c)|<|x-y|M$. Where $f'(x)<M\,\forall\,,x\in\mathbb{R}$. $M>0$

Then you have for any $\varepsilon>0$ if you pick $\delta=\frac{\varepsilon}{M}$. You will have uniform continuity.

The Lipschitz Criteria is that there exist some $M>0$ such that $|f(x)-f(y)|<M|x-y|$. This directly implies but is not implied by uniform continuity.

So bounded derivative$\implies$ Lipschitz$\implies$ uniform continuity. In general none of these arrows are reversible.

And no there is no such trick for sequences of real numbers and neither sequence of functions. There exists sequence of functions whose derivatives exist and is bounded but the convergence is not uniform. You need an additional criteria of pointwise convergent for that to hold