Why uniform convergence of a series doesn't preserve boundedness?

Question

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?

Answer 1

For a fixed $n$, $$|f_x(x)|\leq n,$$ so $f_n$ is bounded.

Uniform convergence preserves boundedness because if $f_n\to f$ uniformly and you advance enough along the sequence, you can find some $f_n$ with $|f_n(x)-f(x)|<1$ for all $x$. Then $$ |f(x)|<|f_n(x)|+1\leq M_n+1 $$ (where $M_n$ is a constant such that $|f_n(x)|<M_n$ for all $x$), and so $f$ is bounded.

And for your last question: yes. Uniform convergence allows you to basically "forget" about the tail of the series and work on finite sums.