I've been told to note that this is "self-learning" by someone with 29k rep, this is self learning!
"If we assume $f_n:[a,b]\rightarrow\mathbb{R}$ and $f_n(x)=(1-x/n)^n$ converges uniformly we can find $f'_n/f_n:[a,b]\rightarrow\mathbb{R}$ which converges uniformly to -1 from this one can show $f'_n$ converges uniformly to $-f$ (yes -f) Thus we can state $f'=-f$
use this to state a theorem about the derivative of the limit of convergence sequences to $C^1$ functions in exercise 2"
Unfortunately there are no even numbered solutions in the back, and I have no clue what it is on about, I hate to say that but really, what!?
I don't like how it starts with "if we assume" and I don't like going from an example I don't trust to a theorem, what is it trying to get me to see? Is it even right?
(I'm scribbling away on paper right now, I'll edit-in an addendum if I find anything meaningful - I've been pondering this for 2 days now and gotten no where)
The book is "Mathematical analysis: a fundamental and straightforward (ha) approach"
Alec, it's a bit hard to decipher your question. So, let's assume $f_n$ converges uniformly to $f$ on $[a,b]$ (it is true for any $a$, $b$). You can explicitly calculate $\dfrac{f_n'(x)}{f_n(x)}=-\dfrac1{1-x/n}$, and this converges uniformly to $-1$ (why?) on any fixed interval. Now use basic stuff to show that if $f_n'/f_n \to -1$ uniformly, then $f_n'\to -f$ uniformly.
Do we know $f_n'\to f'$? What do we know about this sort of question? (I gather this is what question #2 is referring to in some form?)