As I understand, the general theory of limits of real functions is enough to infer all of the important results of theory of sequence convergence. However, theory of sequence convergence is most commonly presented before.
In this question I want to verify that the reason for this is only pedagogical (and maybe historical) but that in the end the general theory of limits does not requiere it and that it is only presented a as an intruduction to limiting processes
Moreover, I would like to verify that the definition of the limit of a function when its independent variable tends to infinity and its corresponding associated theorems, e.g. that it exists when the function is bounded and monotonic, are enough to trivialize sequence convergence results
The only possible exception is the General Principle of Convergence, for the one I have only found a demonstration that uses Cauchy Sequences. The purpose would be to dispense all use of sequences as they would just be a subset of real fuctions that have a restricted domain (that is, the natural numbers).
This purpose would serve to present limits first, and then treat sequences as a particular simpler case and use their results when strictly necesary, e.g. when truly being limited to a natural numbers domain. This would put enfassis on the independence and completeness of (truly) real analysis methods, because I think we can get benefit from having independent and complete representations of these 2 sets of real analysis methods
For example, it has been shown that algebra of continuous function theorems can be derived from both, algebra of "general" limits theorems and from algebra of limits of sequences (using Sequential Continuity Theorem) Additionaly, this leads to a simmiliar sepparation between series and integrals where series are limits of sequence partial sums and integrals are limits of "function partial sums" (a proper correspondence is out of scope) I think the dichotomy may be deeper from what is seen at first glance
