# Analysis...

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#1
This is just part of extra reading I'm doing for university. An example in the book I'm reading is proving that the cube root of 2 exists.

S = the set of all real numbers x, such that x^3 < 2

Let c = the least upper bound of the set.

The book then goes on to show c^3 = 2, by contradiction. It starts by looking at c^3 < 2. This is where I am confused. The book explains we are trying to find a small number y > 0, where (c + y)^3 < 2

(c + y)^3 < 2 implies c^3 +3c^2y + 3cy^2 + y^3 < 2
Therefore, 2 - c^3 > 3c^2y + 3cy^2 + y^3
Therefore, 2 - c^3 > y(3c^2 + 3c + 1) and 0 < y < 1

The very last step is where I am confused, how have they reached those last two inequalities? I can see you could take a factor of y out, but it doesn't factorise to that. Am I missing something obvious?
0
5 years ago
#2
is this from Prof. Liebeck's book?

Be very careful of the direction of the implication signs. Read it though again:you have written the implications the wrong way round.
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#3
Ah yes, I think I see. Thanks 0
#4
(Original post by arkanm)
Just saw this (please quote me for quicker replies )

I still don't see why we can't just use the continuity (from reals to reals) of y=x^3 to show that a cube with volume 2 exists. Then note that any edge length of such a cube will be cube root of 2. Since the edge length of the cube exists, therefore cube root of 2 must likewise exist.
Well, I suppose we can do that. I wanted to understand the example in the book though, as I've written about the book in my personal statement, and it probably wouldn't look very good if I didn't understand it XD
1
5 years ago
#5
(Original post by arkanm)
Just saw this (please quote me for quicker replies )

I still don't see why we can't just use the continuity (from reals to reals) of y=x^3 to show that a cube with volume 2 exists. Then note that any edge length of such a cube will be cube root of 2. Since the edge length of the cube exists, therefore cube root of 2 must likewise exist.
I suspect you're invoking properties of inverse functions and continuity there. Although your argument can be made valid, it's better to let the OP follow the logic given in the book - analysis is one area where people can get into bad habits if they start assuming things that look 'obvious' 0
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