Need help on the syntax of a basic proof

[jokerthief]

I have two questions on an assignment that require me to write two ε-δ Proofs. I understand the logic behind the ε-δ definition of a limit but I've never been asked to write a proof before and there aren't any examples in our book. I understand the semantics but not the syntax of what I need to do.

One of the questions is:

Give an ε-δ proof that: lim (2x+5) = -1
x-->(-3)

I know from the definition of a limit that if 0<|x+3|<δ, then |2x+6|<ε. After doing some basic algebra, I know that δ = ε/2.

So how do I write a proof with this information? I know that I could probably just put what I know into words and get credit (in fact, my prof told me to do exactly that) but I want to know how to write a tight formal proof.

 

[HallsofIvy]

I have two questions on an assignment that require me to write two ε-δ Proofs. I understand the logic behind the ε-δ definition of a limit but I've never been asked to write a proof before and there aren't any examples in our book. I understand the semantics but not the syntax of what I need to do.

One of the questions is:

Give an ε-δ proof that: lim (2x+5) = -1
x-->(-3)

I know from the definition of a limit that if 0<|x+3|<δ, then |2x+6|<ε. After doing some basic algebra, I know that δ = ε/2.

So how do I write a proof with this information? I know that I could probably just put what I know into words and get credit (in fact, my prof told me to do exactly that) but I want to know how to write a tight formal proof.

Exactly what basic algebra did you do? I hope you didn't throw away your scratch paper! That's crucial to the proof!

I imagine what you did was start with |(2x+5)-(-1)|= |2x+ 6|= 2|x+ 3|< \epsilon so |x+6|= |x-(-3)|< \epsilon/2.

Now reverse that:
Take \delta= \epsilon/2. If |x-(-3)|= |x+ 3|< \delta= \epsilon/2, then 2|x+3|= |2x+ 6|= |2x+5-(-1)|< \epsilon.

Actually, what you did initially is perfectly good and is what you will see in most textbooks. It is sometime called "synthetic" proof. Start from what you want to prove and go to something that is part of the hypothesis). As long as every step is reversible, the actual proof, going form the hypothesis to what you want to prove, is obvious and doesn't have to be stated.

 

[jokerthief]

Exactly what basic algebra did you do? I hope you didn't throw away your scratch paper! That's crucial to the proof!

Ha ha, no, I'm not that much of a noob. I pretty much knew what to do but wasn't sure if I was correct. Your post helped confirm to me that I was going in the right direction. I was trying to make it more difficult than it actually was. Thanks HallsofIvy, your post helped me feel confident turning in my assignment today.