I am new to MSE, so apologies for any inconveniences regarding my question. I am studying for the GRE's Math Subject and decided to go back and study certain important concepts from the Analysis series. I am reading Elementary Analysis by Ross, particularly section 24, Uniform Convergence.
We are shown the sequence of real valued functions
$$f_n(x) = nx^n, \quad x \in [0,1),$$ and told that it does not converge uniformly on said interval. However, $\ f_n \rightarrow f$ point-wise on $[0,1)$ where $\ f(x) = 0, \ \forall x \in [0,1).$
This all makes sense and I am fine with it. However, it is later remarked that a sequence $\ (f_n)$ of functions on a set $\ S \in \mathbb{R}$ converges uniformly if and only if
$$lim[sup\{|f(x)-f_n(x)|:x \in S\}] = 0.$$
This is what confuses me, because I am pretty sure (not $100 \%$ sure), that since $limsup = L = liminf$ for any convergent sequence (where $L$ is the limit), then for the example $f_n(x) = nx^n$, shouldn't this sequence of functions technically be uniformly convergent?
I'm most definitely missing something, though hopefully not anything glaringly obvious, and would appreciate some feedback. Thanks!
Best Answer
For bounded functions $f:S\to \Bbb R$ and $f_n:S\to \Bbb R$ let $$\|f-f_n\|=\sup_{x\in S}|f(x)-f_n(x)|.$$ Consider $\|f-f_n\|$ to be a measurement of how closely $f_n$ approximates $f,$ not at any one $x\in S,$ but as an over-all measurement over all of $S.$ Consider $f$ and $f_n$ as single objects, and $\|f-f_n\|$ as the distance from $f$ to $f_n.$
The sequence $(f_n)_n$ converges uniformly iff $\|f-f_n\|\to 0$ as $n\to \infty.$
If $(f_n)_n$ converges uniformly to $f$ then for any $y\in S$ we have $$|f(y)-f_n(x)|\leq \|f-f_n\|,$$ so $f_n(y)\to f(y)$ as $n\to \infty.$
If $f_n(y)\to f(y)$ for each $y\in S$ we say that $f_n$ converges point-wise to $f.$ So uniform convergence implies point-wise convergence (but not vice-versa). If we are testing whether $(f_n)_n$ converges uniformly to some (any) $f$, we can, as a first step, identify the "candidate" $f$ by evaluating $\lim_{n\to \infty}f_n(y)$ for each $y\in S.$
With $S=[0,1)$ and $f_n(x)=nx^n,$ then $f_n(y)\to 0$ for each $y\in S.$ By the previous paragraph, if $(f_n)_n$ did converge uniformly to a function $f,$ then we would have $f(y)=\lim_{n\to \infty}f_n(x)=0$ for each $y\in S$..... But for any $n\in \Bbb N$ we can find some $y_n \in S$ with $y_n$ close enough to $1$ that $(y_n)^n>1/2.$ Then $f(y_n)>n/2$ and $f(y)=0$ so $$\|f-f_n\|\geq |f(y_n)-f_n(y_n)|=|0-f_n(y_n)|>n/2.$$ So the sequence $(f_n)_n$ in the Q is not a uniformly convergent sequence..
BTW . An important result that holds for all metric spaces $S, T$ (not just sub-spaces of $\Bbb R$ ), is that if $(f_n:S\to T)_{n\in \Bbb N}$ is a sequence of continuous functions converging uniformly to $f:S\to T$, then $f$ is continuous.