Proving Simple Inequality: a < b + c < b + d | Homework Solution

[brntspawn]

Homework Statement

Prove:
If a < b and c < d then a+c < b+d

Homework Equations

The Attempt at a Solution

Proof
Assume a < b and c < d
then a+c < b+c and b+c < b+d
so a+c < b+c < b+d
therefore a+c < b+d
Q.E.D.


Pointers and suggestions are also welcome. I am looking for writing and presentation tips just as much as I am for accuracy.

 

[Squeezebox]

Looks good. The only thing that I would change is "Assume a < b and c < d". Assuming something does not mean it is necessarily correct as a proof by contradiction can demonstrate. So you can change it to something like "Stated a < b and c < d." Since a < b and c < d was given to you.

 

[Mark44]

Looks good. The only thing that I would change is "Assume a < b and c < d". Assuming something does not mean it is necessarily correct as a proof by contradiction can demonstrate. So you can change it to something like "Stated a < b and c < d." Since a < b and c < d was given to you.

Squeezebox, I don't think you know what you're talking about. It is perfectly reasonable to assume that a < b and c < d in this problem, because this statement is the hypothesis.

When you set out to prove a statement such as if p then q, in a direct proof you assume that p is true and then show that this assumption leads to concluding that q is true as well.

 

[brntspawn]

Thanks guys. I was actually wondering about the assume part because I know it was drilled into us in class that we should specifically state it. I assume at some point, perhaps when you move from undergrad classes to grad classes you do not specifically need to state it, but I am not sure.
Are we just supposed to list our assumptions for a proof like this so the teacher knows that we know what we are proving?

 

[Tedjn]

There's nothing wrong with listing your assumptions explicitly. My own style on homework is to write my solution as a paragraph, stating the assumptions when they are needed. In your case, there is only one assumption, which you used in the beginning, so it seems fine to me. Along the way, you should develop your own writing style, and as long as it remains clear, you're free to experiment.

 

[brntspawn]

Yeah that is pretty much where I am at now, realizing that there is no one correct way to write a proof and finding my own style for mine when I write them.
My school used Stewart for its calculus text and I came away with it feeling as though I didn't really understand much. I hated Calculus and couldn't wait to get it over with because all we were doing was doing was plug and chug math. Then I was fortunate enough to have had an instructor for Multivariable Calculus who actually went over some theory and tried to explain what was going on more than "here is this formula, memorize it." So I decided to go through Spivak's Calculus book and do double duty by actually learning Calculus and finding my own proof writing style.