Prove that the function is of exponential order and proving in mathematics

Question

I'm currently learning about the Laplace transform and in my textbook in college and I have this definiton:

Function $f$ is of exponential order if there exist constants $M>0$ and $a$ such that for every $t>0$ relation $|f(t)| \le Me^{at}$ holds.

I have the folloiwing assignment: let the function $f$ be continuous on $[0, \infty >$, prove that it is of exponential order when and only when there exist constants $a, M$ and a number $t_{0} > 0$ such that $|f(t)| \le Me^{at}$, for every $t > t_{0}$ holds.

How do I prove (go about proving) this? How do you usually go about proving something? Do proofs only need to contain math symbols or also some text? How to practice proving mathematical statements?

Answer 1

We need to prove 2 things:

• if a function is of exponential order then there exist constants...,
• if there exist constants... then the function is of exponential order.

If there are constants $a,M$ and $t_0>0$ such that $|f(t)|\leq Me^{at}$ for every $t>t_0$, then in particular $t>0$, because $t>t_0>0$. Therefore the function is of exponential order (by definition).

If a function is of exponential order then (by definition) $|f(t)|\leq Me^{at}(*)$ for every $M>0,a,t>0$. In particular if $t_0>0$ then $(*)$ holds for every $t>t_0$.

You need to identify what you want to prove. In this case it is a biconditional. If you want to practice proving mathematical statements I recommend you to take a course on foundations of maths or read a book like Proofs and Fundamentals by Bloch. Euclidean geometry may also help you to understand proofs and write to them in a proper way.

Answer 2

Hint: For any real $b$ we have $K(b)=\min \{e^{bx}: x\in [0,t_0]\}>0$, and since $f$ is continuous,we have $m(f)=\max \{|f(x)|:x\in [0,t_0]\}<\infty.$

So for all $x\in [0,t_0]$ and all $b\in \mathbb R$ we have $|f(x)|/e^{bx}\leq m(f)/K(b)<\infty.$

The idea is that if $|f(x)|\leq Me^{ax}$ for all $x>t_0> 0,$ we can find $M'\geq M$ and $b\geq |a|$ such that $|f(x)|\leq M'e^{bx}$ for all $x\geq 0.$