# Why the following description implies the limit is finite?

Let $$f:[0,T]\rightarrow \mathbb{R}$$ $$V(f) = \lim_{\|\Pi\|\rightarrow 0}\sum_{j=0}^{n-1}|f(t_{j+1})-f(t_j)|$$

where $$\Pi$$ is a partition of $$[0,T]$$ and define $$\|\Pi\|$$ as

$$\Pi=\{t_0,t_1,\cdots,t_n\}, \ \ 0=t_0

Now the solution manual says that

Suppose $$V(f)$$ is finite. Then for any $$\epsilon>0$$, there exists an $$N\geq 1$$, $$\sum_{j=0}^{n-1}|f(t_{j+1})-f(t_j)| for all $$n\geq N$$.

I am confused about the description of $$+\epsilon$$ and $$\forall n \geq N$$ parts. How do they and this description imply finite of $$V(f)$$?

I believe this is a trick widely used in real analysis; however, cannot still understand this.

It just follows from the definition of $$V(f)$$ as a limit.

Given $$\epsilon>0,$$ there is a $$\delta>0$$ such that for $$any$$ paritition $$P=\{0,x_1,\cdots ,x_{n-2},T\}$$ of mesh less than $$\delta$$, we have

$$\left|\sum_{j=0}^{n-1} |f(t_{j+1}) - f(t_j)|-V(f)\right|<\epsilon\tag1$$

So, choose a partition $$P$$ with $$N-1$$ large enough so that the mesh of $$P$$ is less than $$\delta.$$ Then, $$(1)$$ holds, from which it follows that

$$-\epsilon+V(f)<\sum_{j=0}^{N-1} |f(t_{j+1}) - f(t_j)|

But any partition with $$n>N$$ that refines $$P$$ will have mesh less than $$\delta$$ so in fact $$(2)$$ holds for $$\textit{all}\ n\ge N$$ and this is exactly the result you have in blockquotes.

If $$a_n$$ is a sequence whose limit exists (call it $$a := \lim_{n \to \infty} a_n$$), then by the definition of a limit, for each $$\epsilon > 0$$ there exists $$N$$ such that $$|a_n - a| < \epsilon$$ for all $$n \ge N$$. Note that $$|a_n - a| < \epsilon$$ implies $$a_n < a + \epsilon$$.

Your situation is pretty much the same, except the limit is taken as $$\|\Pi\| \to 0$$. So for any $$\epsilon > 0$$, there exists a $$\delta$$ such that $$\sum_{j=0}^{n-1} |f(t_{j+1}) - f(t_j)| < V(f) + \epsilon$$ for any partition satisfying $$\|\Pi\| < \delta$$. Presumably you have omitted some condition (e.g., you are considering evenly-spaced partitions) that would imply that partitions $$\Pi$$ of size $$\ge N$$ will have small $$\|\Pi\|$$.

What are you asking for the converse of what is sated in the manual. The manual says that if $$V(f)<\infty$$ the something happens. The stated property is immediate from the definition of limit. It is something like this: if $$\lim a_n=l$$ and $$\epsilon >0$$ then there exists $$N$$ such that $$a_n for $$n \geq N$$. Do you agree with this statement?