A Is This Proof Correct? Ask & Discuss Here

[JasMath33]

I was reading this book yesterday and looking at this proof/justification. I was thinking it is possibly incorrect, but wanted to get some other opinions. Here is the example they gave in the book with the work attached.

 

[fresh_42]

Looks ok to me. What does make you think it might be incorrect?

 

[JasMath33]

Looks ok to me. What does make you think it might be incorrect?

I guess what threw me off was the ending. I was just expecting to see something stating therefore the the function is Lipschitz. I guess in the book, they wanted you to just in your head think that.

 

[fresh_42]

I guess what threw me off was the ending. I was just expecting to see something stating therefore the the function is Lipschitz. I guess in the book, they wanted you to just in your head think that.

Well, it's the definition of Lipschitz continuity. If one has to show a number $n$ is even and shows it is divisible by $2$, nobody would complain about a missing "... therefore $n$ is even", because we are used to the concept of even numbers. It is the same here: simply a matter of acquaintance. Maybe it would be a little more obvious if one had chosen $L$ instead of $M$ as the constant.

 

[JasMath33]

Well, it's the definition of Lipschitz continuity. If one has to show a number $n$ is even and shows it is divisible by $2$, nobody would complain about a missing "... therefore $n$ is even", because we are used to the concept of even numbers. It is the same here: simply a matter of acquaintance. Maybe it would be a little more obvious if one had chosen $L$ instead of $M$ as the constant.

Yeah I see what you are saying now. It makes sense. Thanks.