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 
$$\Pi=\{t_0,t_1,\cdots,t_n\}, \ \ 0=t_0<t_1<\ldots<t_n=T, \ \ \|\Pi\|=\max_i(t_{i+1}-t_i)$$
Now the solution manual says that 

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

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 $a_n <l+\epsilon$ for $n \geq N$. Do you agree with this statement? 
