Prove that the function is of exponential order and proving in mathematics 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? 
 A: 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 $x\geq 0.$
A: If the function is of exponential order, then by definition it follows that we can find constants $a$ and $M>0$ such that $$|f(t-t_0)|\le  Me^{a(t-t_0)}$$ whenever $t>t_0.$ Since this function is continuous it follows that as $t_0\to 0,$ we must have $f(t-t_0)\to f(t).$ Therefore, in this limit, we have $$|f(t)|\le  Me^{at},$$ since the exponential function is continuous, for every $t>t_0,$ as wanted.
On the other hand if we have that for some appropriate constants $a,M$ the inequality $|f(t)|\le  Me^{at}$ holds for all $t>t_0,$ then it follows that by setting $t-t_0=s>0$ that we have $|f(s+t_0)|\le  Me^{a(s+t_0)},$ and again since $f$ is continuous, it follows that in the limit as $t_0\to 0,$ we have that $$|f(s)|\le  Me^{as}$$ for $s>0$ and suitable constants $a,M,$ and now the proof is complete.
For your question about proofs in mathematics, there is no general method. But in this case as in many others, it's just a question of applying the given definitions and hypotheses.
A: 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.
