# Principles Of Mathematical Analysis-Walter Rudin

## Main Question or Discussion Point

I want to discuss about Chapter 8, equation 43 for any real number α(alpha)

The author tells "The continuity and monotonocity of E and L show that this definition
leads to the same result as the previously suggested one".(i.e.,Equ 33)

But I think properties of L is not required for the proof and proof is the same as the proof
for equivalence of equ 34 and 35.

I would like to get an expert comment on this.

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Nobody replied to my query. Any way I will put my proof which doesn't use the properties of L. But I think the
proof must be wrong.

Fix any real k > 1.
f(x) = sup( k raised to y) (y rational , y<x)
g(x) = E(xL(k))
Now we have to prove that f(x) = g(x) for any real x.
For rational x, we agree that f(x) = g(x).

Now suppose f(x) < g(x). Because of continuity and monotonicity of g(x) (in x) we can find a d such that f(x) < g(p) < g(x) for any p in x-d < p < x. Now pick any rational p then f(x) < g(p) = f(p) < g(x) which is impossible since f(x) is an upper bound and hence f(x) > f(p).

Now suppose g(x) < f(x). Now we can find rational p < x such that g(x) < f(p) < f(x).
But then f(p) = g(p) > g(x) which is impossible since g is monotonically increasing function.

Here I didn’t use any monotonicity and continuity of L!

I want to discuss about Chapter 8, equation 43 for any real number α(alpha)

The author tells "The continuity and monotonocity of E and L show that this definition
leads to the same result as the previously suggested one".(i.e.,Equ 33)

But I think properties of L is not required for the proof and proof is the same as the proof
for equivalence of equ 34 and 35.

I would like to get an expert comment on this.
We need more information if you want us to help you. Not all of us want to pull out our copy of Rudin just to look for your problem.

But how to provide more information? I have soft copy of the book but its size is 12MB, so I cannot upload it.
Maybe I will have to purchase a pdf editor?

I have so many comploicated questions in the book. I am confused how to ask it.

But how to provide more information? I have soft copy of the book but its size is 12MB, so I cannot upload it.
Maybe I will have to purchase a pdf editor?

I have so many comploicated questions in the book. I am confused how to ask it.
Giving context is a good start. What are ##E## and ##L##?

E&L

E is defined by equation 25 chapter 8 page no 178
i.e., E(z) = Sigma(n=0 to INF)( (z raised to n)/(n!))
and L is inverse of that.Why the inverse exists, why E(z) is continuous and monotonically increasing like that are proved in the book.

Finally the author defines x raised to y as E(yL(x)) for all real x and y.

What I want to prove is that this is equivalent to x raised to y = sup (x raised to r)(r rational, r < y) for x > 1. But the author tells monotonicity and continuity of L is required for the proof, which I didn't use in the proof I gave.

Please go through the book. Now I am struggling with Theorem 8.14. I will post the query as a
separate question.

Shaji is right, without more info, nobody will pull out the book and answer you. And it helps if you know how to typeset in LaTeX, it took me quite a while to understand your E and L.

I know ehere the continuity is needed, inyour statement, you said
"Now suppose g(x) < f(x). Now we can find rational p < x such that g(x) < f(p) < f(x). "

That is the "Intermediate Value Theorem" that requires continuity.

No. We are not using continuity of any function here. If there is no rational p < x with g(x) < f(p) < f(x), then
f(x) is not the supremum. supremum will be < or = g(x).Note that f(x) is defined as f(x) = sup( k raised to y) (y rational , y<x)(k>1).

Also continuity of L(k) is the issue. In the proof we keep k as a constant. Note that g(x) = E(xL(k)).
Also here we consider the case k > 1 only.

If you have the book and would otherwise willingly spend probably several minutes to write an answer, why would you refuse to take the extra 30 seconds to find eq.43 in Chapter 8?

If you wanted the OP to show more work, that's another thing of course.

I am asking the question honestly.

Equ.43 is proved for any rational number and taken as definition for any real number and author claims that
the definition is equivalent to equ.33 chapter 8(which is another definition). Author claims continuity and
monotonicity of L is required for the proof of equivalence which I dispute.

Mark44
Mentor
Questions from textbooks should be posted in the Homework & Coursework section, not in this section. When you post a problem, include the theorem that you are trying to prove and the work you have done. Not all of us have a copy of Rudin, so references to specific problems are meaningless to us.

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