# How to determine whether a sequence of functions converges uniformly or pointwise to a function?

How to determine whether a sequence of $\mathbb{R}$ functions converges uniformly or pointwise to a function? Although I know the definition, I don't fully understand how to use it. Does this correlate to the derivative of a function? I.e., my thinking is along the lines of, if a function grows two quickly, limsup of the error, as n->inf, on all of the domain won't be 0. To understand this concept, is to be able to come up with functions satisfying the required domain and convergence criteria. Therefore, if you could do such an example it would probably help me understand. For example, how do I determine the type of convergence of $lim_{n\rightarrow\infty} x^n=0$ on the interval [0,1)?

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## 1 Answer

You need to go back to the definitions of pointwise convergence and uniform convergence and prove the point in each case. But there can be indications.

In your example, one way to spot that there might be a difference is that $x^n$ is continuous on the slightly wider interval $[0,1]$, but that $1^n$ does not converge to $0$.

If you had been looking at the function $n^2(1-x)x^n$ then an indication would be that the integral over the interval does not converge to $0$ so there will not be uniform convergence even if there is pointwise convergence.

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