I didn't add a lot of context so I just edited it.
If I have a sequence of functions $ \{f_n\} $ which are continuous on $[a, b]$, and if they converge uniformly on every subset of $(a, b)$: $\forall S \subsetneq (a, b)$, do they converge uniformly on $(a, b)$? and thus implying uniform convergence on $[a, b]$?
I think it is false since, assuming it is true then we can look at the following function: $$\sum_{n=2}^{\infty} \frac{\sin(nx)}{\ln(n)}$$we can then look at the sequence of functions $ f_N(x)=$$\sum_{n=2}^{N} \frac{\sin(nx)}{\ln(n)}$$ $since they are all continuous on [$0$,$2\pi$]and I alreaedy proved using dirichlet criterion that they converge uniformly for every $0$<$\delta$<$\pi$ on [$\delta$,$2\pi$-$\delta$]. thus assuming it is true then they converge uniformly on (a,b) and thus on [a,b]. such a thing would imply that the fourier series of this function( the function which they converge) would bet this function itself namely $$f(x)=\sum_{n=2}^{\infty} \frac{\sin(nx)}{\ln(n)}$$which is a contradiction knowing that this infinite sum is not a fourier series of any function
