Fn uniformly converges to f: for any e0, there is a N0, so that for any X, it is in the definition domain and NN, | f (x)-fn (x) | < e.

Fn converges to f point by point: for any e0, for any X in the domain, there is an N_x0, so that any sum of n _ x, | f (x)-fn (x) | < e.

Note here that I put a subscript x on n that converges point by point, indicating that n and x are related. The biggest difference is that the uniformly convergent n is applied to all x:

Point-by-point convergence means that the function value fn(x) converges to f(x) at each point, but the convergence speed may be different at different points.

Uniform convergence means that all fn(x) converge to the whole f(x) approximately "synchronously". Q:

Does convergence necessarily mean uniform convergence? Answer:

Not necessarily, but on the contrary! Q:

I have seen a series of continuous answer proof questions, first prove its convergence, then directly reach uniform convergence and then directly reach continuity ... I don't quite understand ... Answer:

Why introduce the concepts of consistency and continuity? It is easy to prove that the sum of finite "continuous functions" is still "continuous functions". But for the series of function terms, each function is continuous, the series converges to each point X, and the series of function terms converges to the sum function S(x). It is natural to ask whether S(x) is a continuous function. Sorry, I can give a counterexample, not necessarily. There is the following theorem: on the closed interval, every term in the series of function terms is continuous and uniformly converges to S(x), then S(x) is also continuous on the closed interval. In the above discussion, we paid attention to "continuous function" and changed it to "derivative function", and "integrable function" was also established. It can be seen how powerful the concept of uniform convergence is! What needs to be added is that you said that what you saw was that you got uniform convergence after proving convergence. You can look up several judgment theorems of uniform convergence in the analysis book, one of which must have been used in the proof.