In my notes, there is this theorem:
A sequence in $R^n$ converges (to a limit in $R^n$) iff it is Cauchy.
I understand that this theorem applies to all complete metric spaces, not just to $R^n$.
Now, here is my question:
The sequence of continuous functions $(f_n)$ in the space of continuous functions $C[0,1]$ equipped with the metric
$$d(f,g)=\sup_{x\in[0,1]}|f(x)-g(x)|$$
also converges to a limit in $C[0,1]$ iff it is Cauchy since $C[0,1]$ is complete.
But, what the convergence here explicitly is? Is it pointwise convergence or uniform convergence?
I understand that uniform convergence implies pointwise convergence with the same limit. Is it possible in the case above that there is a sequence of functions that converges pointwise and Cauchy in $C[0,1]$ but does not converge uniformly?
So is it more appropriate is it to re-write the Theorem as a sequence of continuous function in converges uniformly in $C$ iff it is Cauchy?
Thank you for the clarification.
Best Answer
In the space of continuous function if you take any sequence $\{f_n\}$ with respect to any metric $d(f,g)$, if it is uniformly bounded and equi-continuous then you can say it converges uniformly by Arzela-Ascoli Theorem.