analysis - What are common methods/techniques can be used to prove that limit of an infinite sequence exists?

I would like to know what are common methods can be used to show that an infinite sequence converges. From what I know so far,

1. If a sequence is bounded and monotonic increasing/decreasing then it converges.


2. Using definition of limit.


3. Another method that I saw online, is to assume the sequence approaches a limit $L$, then solve for $L$, but I'm not totally convinced that this approach is correct. For example, the Fibonacci ratio sequence, to prove the limit of $$\displaystyle\lim_{n\to\infty} \dfrac{a_{n+1}}{a_n}$$ exists, they claim that:
   $$1 + \dfrac{1}{L} = L$$
   proof for 3

So I wonder could anyone could share me some most commonly used method for proving the limit of an infinite sequence exists that I'm not aware of? Any suggestion or ideas would be appreciated.

Answer

One of the most powerful tools in calculus(but not only):

Cauchy's criterion for convergence