I am revising my analysis notes from last year for a course this year...and I am very not sure about pointwise/uniform convergence.
I know there are lots of answers around, as it is one of the most confounding ideas and definitions in math which probably almost no one got it completely at a first glance.
I am looking at a specific question which, the solution, doesn't make sense to me.
Q: $f_n=\frac{x \sqrt{n}}{1+nx^2}$ for $x \in \mathbb{R}$. Prove that $f_n$ converges pointwise to the $0$ fnction. Is the convergence uniform over $\mathbb{R}$?
Well, now here is the solution,
A:If $x=0$ then $f(0)=0$ so it converges to $0$. If $x \neq 0$, $|f_n(x)| \leq \frac{|x| \sqrt{n}}{nx^2}= \frac{1} {\sqrt{n}|x|}$ thus as $n$ tends to infinity, it converges to $0$ pointwise. Take $x= \frac{1}{\sqrt{n}}$, then $f_n(\frac{1}{\sqrt{n}})= \frac{1}{2}$ therefore not uniform over $\mathbb{R}$
Now, I know many will suggest I go back to the definitions of each convergence but here, I want to understand them through this specific question; I know pointwise is dependent on both $x$ and $\epsilon$ while uniform is only dependent on $\epsilon$.
But here, I just can't see the logic; we have proven that for ALL $x \neq 0$, the sequence of function converges to $0$. I don't see why this is only "pointwise" because we have argued with some general $x$ that is NOT $0$ and the result is that $f_n$ apraoches $0$ along with increasing $n$.
Then it specifies $x = \frac{1}{\sqrt{n}}$ and inputs that to the function and then we suddenly get $ \frac{1}{2}$. But, we just said that for an $x$ which is not $0$ (well, also for 0) will converge to $0$. Why does this particular $x$ go to $\frac{1}{2}$? Not in the literal sense of course, I mean, a simple calculation indeed tells me it is $\frac{1}{2}$ but I cannot see why this is valid while the argument before is also valid, that $x \neq 0$ leads to convergence to $0$.
It's like a package of contradiction to me; Staring at the definitions for hours didn't help, so please, someone explain to me using this example please??
Thank you so much!