There is a series of functions $f_n:(0,1) \rightarrow R$ defined as $f_n(x)=\frac{n}{nx+1}$.
The limit of the series is $f(x) = \lim_{n\rightarrow \infty}f_n(x) = \lim_{n\rightarrow \infty}\frac{1}{x+\frac{1}{n}} = \frac{1}{x}$. So $f_n\rightarrow f$ pointwise on $(0,1)$.
Could you please explain:
1) $f$ isn't bounded on $(0,1)$, because $\lim_{x\rightarrow0} \frac{1}{x} = \infty$. Why are $f_n$ bounded on $(0,1)$ (should we consider $n$ strictly less than $\infty$?
2) Why in general pointwise convergence doesn't preserve boundedness, but uniform convergence does? Is the preservation of boundedness by uniform convergence the reason for the possibility of interchanging limits of uniformly convergent series?