Prove that
if $f$ is of bounded variation in $[a,b]$, it is the difference of two positive, monotonic increasing functions; and
the difference of two bounded monotonic increasing functions is a function of bounded variation.
In fact, I believe it should not be too difficult to show $\operatorname{Var}[a,x_2] = \operatorname{Var}[a,x_1]+ \operatorname{Var}[x_1,x_2]$. Which is slightly stronger than the inequality in the answer.
The basic idea to show that the supremum over those special partitions is indeed $\operatorname{Var}[a,t_1]+|f(t_2)-f(t_1)|$ is noticing that $$\sup \left(|f(t_2)-f(t_1)| +\sum_{i=1}^{n-1} |f(x_n)-f(x_{n-1})|\right)= |f(t_2)-f(t_1)| + \sup \left(\sum_{i=1}^{n-1} |f(x_n)-f(x_{n-1})|\right)$$
Suppose That three points on the graph of $y=x^2$ have the property that their normal lines intersect at a common point. Show that the sum of their $x$-coordinates is $0$.
I have a lot going but can not finish it.
Proof:
Let $(a,a^2)$, $(b,b^2)$, and $(c,c^2)$ be three distinct points on $...
It should be relatively easy to find a counterexample. In fact, I think any function which is not monotone on the interval $[t_1,t_2]$ will give you a counterexample.
@BAYMAX No, it does not. What I am trying to prove is to get in the end this inequality: $$\operatorname{Var}[a,t_1]+|f(t_2)-f(t_1)|\le\operatorname{Var}[a,t_2].$$
The equation $(*)$ in combination with what I wrote before gives use the desired inequality: $$\operatorname{Var}[a,t_1]+|f(t_2)-f(t_1)| \le \operatorname{Var}[a,t_2]$$
The basic idea to show that the supremum over those special partitions is indeed $\operatorname{Var}[a,t_1]+|f(t_2)-f(t_1)|$ is noticing that $$\sup \left(|f(t_2)-f(t_1)| +\sum_{i=1}^{n-1} |f(x_n)-f(x_{n-1})|\right)= |f(t_2)-f(t_1)| + \sup \left(\sum_{i=1}^{n-1} |f(x_n)-f(x_{n-1})|\right)$$
I do not think we have used this anywhere. (And I am not really sure what you mean by this.)
BTW there is an obvious typo in my message above.
It should have been $$\sup \left(|f(t_2)-f(t_1)| +\sum_{i=1}^{n-1} |f(x_i)-f(x_{i-1})|\right)= |f(t_2)-f(t_1)| + \sup \left(\sum_{i=1}^{n-1} |f(x_i)-f(x_{i-1})|\right)$$
To related it to the above inequality, $|f(t_2)-f(t_1)$ is the constant. (It does not depend on which partition belonging to $\mathcal P_1$ we choose.)
So there are instances where i can find an explicit expression for the function mapping natural numbers to a set which we would like to show that it is countable.
And there are also instances where i cannot find an explicit expression for the function, then i would suggest a method/algorithm to map the natural numbers to my set
sometimes, i show it for first 10 natural numbers, then i would write repeating the process for the other natural numbers
Now, the big issue i have is to show that the mapping/function is indeed bijective.
In cases, where i do not have the explicit formula for the function, how do i show that it is a bijection?
For example, the well known case for Rational numbers, the mapping has no explicit form, instead a counting algorithm is proposed.
Prove that
if $f$ is of bounded variation in $[a,b]$, it is the difference of two positive, monotonic increasing functions; and
the difference of two bounded monotonic increasing functions is a function of bounded variation.
Suppose That three points on the graph of $y=x^2$ have the property that their normal lines intersect at a common point. Show that the sum of their $x$-coordinates is $0$.
I have a lot going but can not finish it.
Proof:
Let $(a,a^2)$, $(b,b^2)$, and $(c,c^2)$ be three distinct points on $...
