In my analysis course we are covering the topic of bounded variation fuctions and I am really having a very hard time trying to get the concept.
My main problem is that I don't get how can a function have infinity variation if it is a finite sum! Maybe is because is the suppremum? Still, I don't get it, because we are considering just finite sums, how can it be infinity?
So, the teacher made this classic example:
$f: [0,1] \rightarrow \mathbb{R}$ such that $f(0)=0$ and $f(x)=x\cos(1/x)$ for $x \neq 0$. We got for certain partition:
$$\sum_{i=1}^n |f(x_i)-f(x_{i-1})|> \sum_{i=1}^n 1/2\pi i$$ if we let $n \rightarrow \infty$ the series on the right diverges. I don't get why would we make $n \rightarrow \infty$ if by definition a partition is finite, no matter how big $n$ is (no matter how many points we take in the partition) it's still a number...It may be terrible big, but still, a number...
Probably we need to go back to understand what do we mean by limit.
Let us start with a trivial question. What is this limit? $$ \lim_{m\to \infty} m =?$$ Clearly the answer is $+\infty$. Next, by definition of nature numbers, each $m$ is a finite number, no matter how big it is, it's still a number, however, the limit of $m$ is $\infty$ as $m$ goes larger and larger. Now, compare this example with the example provided by your teacher and think each $m$ is you partition $P$, although each $m$ is finite (each partition is finite sum), but result of limit can be $\infty$. (If the sequence is increasing, then the supremum of this sequence is the limit of this sequence)