Let $\displaystyle f$ be a funtion from $\displaystyle R^+$ to $\displaystyle R$ such that $\displaystyle f$ is of bounded variation on $\displaystyle [0,n]$ where $\displaystyle n \in Z^+$. Let $\displaystyle V_f (0,n)$ be the total variation of $\displaystyle f$ on $\displaystyle [0,n]$. In addition, $\displaystyle f$ satisfies $\displaystyle \mid f(n)-f(0)\mid \leq M_n$ and $\displaystyle \sum_{n=1}^\infty M_n \leq \infty$.
Determine whether or not $\displaystyle lim_{n\rightarrow\infty} V_f (0,n)$ exists.
Let $\displaystyle f$ be a funtion from $\displaystyle R^+$ to $\displaystyle R$ such that $\displaystyle f$ is of bounded variation on $\displaystyle [0,n]$ where $\displaystyle n \in Z^+$. Let $\displaystyle V_f (0,n)$ be the total variation of $\displaystyle f$ on $\displaystyle [0,n]$. In addition, $\displaystyle f$ satisfies $\displaystyle \mid f(n)-f(0)\mid \leq M_n$ and $\displaystyle \sum_{n=1}^\infty M_n \leq \infty$.
Determine whether or not $\displaystyle lim_{n\rightarrow\infty} V_f (0,n)$ exists.
the limit is not necessarily finite. for example $\displaystyle f(x)=\sin(\pi x)$ satisfies all the conditions in your problem but $\displaystyle V_f(0,n)=\int_0^n |f'(x)| \ dx = \int_0^{n \pi} |\cos x| \ dx=2n.$