# Trouble with Rigorous Proofs

## Main Question or Discussion Point

Looking for a little advice regarding proving things in mathematical way. I am a physics major currently taking a math methods course where we are asked to prove things, basically for the time in my schooling career.

Sometimes I have trouble formulating a mathematically rigorous way of putting a proof even if I seem to understand the concept and can explain it in words. To demonstrate what I mean here is an example:

Let f(x) and g(x) be two continuous function on x in [a,b] then prove:

max[ f(x) on [a,b] ] + max[ g(x) on [a,b]] >= max[ [f(x)+g(x)] on [a,b]]

Now I can easily describe in words why this is true, because unless the maximum of f and g coincide the maximum of their sum will be less then the sum of the individual maxima That is either f or g will be smaller than its true maximum in the sum. But I don't really know how to start formulating a nice pretty way of showing it that will satisfy a mathematician.

This is just one example but I tend to go into these sorts of writing arguments a lot on my hw and am worried it is not going to past muster. I'm sure my classmates have a much better grasp on it though so i think I am doing ok in the class, but I'd still like to get better.

THere is no set way to do proofs. Some people write very tidy proofs that are hard for undergrads to follow, but are very efficient in their use of words and expressions.

I personally like to be very wordy and explicit. Using words is NOT A BAD THING in proofs. Its the logic behind your argument that counts.

Usually in rigorous math classes, there is no pre-designed formula for working proofs. You should not also expect to be able to SEE THE PROOF or the way to prove right away; sometimes you do know right away what you need to do, but often you really dont see it.

Having said that, heres what I usually do when the road to a proof is not obvious:

1. Write down every thing I "know" about my structures (by "know" I mean everything that I know through definitions, theorems, propositions, axioms, etc). What do I know about continuous functions? What do I know about adding them over an interval? what do I know about max[]?

2. Write down what it is I am actually trying to prove. What do I have to show in order for my end result to be true? If max [f] + max [g] >= max[f+g], then what else must be true? If thats true what else has to be true, etc etc etc

3. Break your problem down into cases. Suppose max [f] = max [g] what then? What about if max[f] > max [g]?

Chances are once you start thinking your problem in this way, you will probably be able to piece together a good proof.

Again, let me remind you that a "pretty way of showing it that will satisfy a mathematician' is not about using a lot of "cool looking greek symbols" and its not about "using less words" its about having a logically consistent argument built off theorems, props, axioms, definitions, etc. Sometime I prove things and the answer looks more like an essay than a math problem; don't be afraid to use plenty of words.

THere is no set way to do proofs. Some people write very tidy proofs that are hard for undergrads to follow, but are very efficient in their use of words and expressions.

I personally like to be very wordy and explicit. Using words is NOT A BAD THING in proofs. Its the logic behind your argument that counts.

Usually in rigorous math classes, there is no pre-designed formula for working proofs. You should not also expect to be able to SEE THE PROOF or the way to prove right away; sometimes you do know right away what you need to do, but often you really dont see it.

Having said that, heres what I usually do when the road to a proof is not obvious:

1. Write down every thing I "know" about my structures (by "know" I mean everything that I know through definitions, theorems, propositions, axioms, etc). What do I know about continuous functions? What do I know about adding them over an interval? what do I know about max[]?

2. Write down what it is I am actually trying to prove. What do I have to show in order for my end result to be true? If max [f] + max [g] >= max[f+g], then what else must be true? If thats true what else has to be true, etc etc etc

3. Break your problem down into cases. Suppose max [f] = max [g] what then? What about if max[f] > max [g]?

Chances are once you start thinking your problem in this way, you will probably be able to piece together a good proof.

Again, let me remind you that a "pretty way of showing it that will satisfy a mathematician' is not about using a lot of "cool looking greek symbols" and its not about "using less words" its about having a logically consistent argument built off theorems, props, axioms, definitions, etc. Sometime I prove things and the answer looks more like an essay than a math problem; don't be afraid to use plenty of words.
Thanks that was a very helpful response.