Limit of (x^n + y^n)^(m/n) n->infinity Proof

[BrianMath]

Homework Statement

Show that
$$\lim_{n\to \infty} (x^n + y^n)^{\frac{m}{n}} = (max\{x,y\})^m\;\;\;\forall x,y > 0$$
where $max\{x,y\}$ outputs the greater of the two.

Homework Equations

$$\lim_{x\to x_0} (f(x))^n = (\lim_{x\to x_0} f(x))^n$$

The Attempt at a Solution

It's attached as a pdf file. I think I have the proof right, I would just like to know your opinions on how it is laid out. Is it too wordy, or too informal? I'm still not used to writing proofs in a way to convey information to other people, and I've heard that the only way to pick it up is through practice, so that's what I'm doing.

 

[AdrianZ]

well, your proof is correct and convincing, and I don't think that it is too informal, you've just explained every step carefully in details which is good, that doesn't make it wordy or informal. you can avoid going into such details in your exam paper. you have to consider the case when x=0 separately if you want it to be more rigorous. (you're not allowed to factor out x^n if it's zero)

 

[micromass]

You're missing the case where x=y

 

[BrianMath]

You're missing the case where x=y

Ah, yes, thank you.
I've taken care of that, how is it now?


The way I like to write my proofs is to explain every step in detail so that even someone with a basic knowledge of calculus could understand. I think it's mostly to convince myself that I know what I'm doing, but I also want to make sure when someone grades what I do that I don't get points taken away for not providing enough steps or justification.

 

[micromass]

Ah, yes, thank you.
I've taken care of that, how is it now?

Now it's all fine!

The way I like to write my proofs is to explain every step in detail so that even someone with a basic knowledge of calculus could understand. I think it's mostly to convince myself that I know what I'm doing, but I also want to make sure when someone grades what I do that I don't get points taken away for not providing enough steps or justification.

I think your proof was quite beautiful. If they showed it to me, then I would say that it's from a textbook. I like how you explained what you're doing, instead of just writing the calculation...
Keep writing proofs like this!

 

[BrianMath]

Now it's all fine!
I think your proof was quite beautiful. If they showed it to me, then I would say that it's from a textbook. I like how you explained what you're doing, instead of just writing the calculation...
Keep writing proofs like this!

Thank you for the compliment! I only ever see proofs that way in textbooks, so I was worried that I might be writing a bit too textbook-stylish (but now I see that's a good thing). I'm self taught for the most part (since I was 13), so I know when I do computations right, but the kind of ambiguity involved in the art of proof writing always made me wary of whatever I write.

Now I'm confident in my proof-writing abilities.