Help solving problem involving inequalities

[mlearner]

Hi all,

I'm trying to teach myself analysis using the book "A Companion to Analysis: A Second First and First Second Course in Analysis" by T. W. Körner.

There's an inequality problem in there that's used to prove a statement about the continuity of a function, that I've got stuck in (problem 1.16, part (iii), in case you happen to have the book):

Working in \mathbb{Q} (the space of rational numbers), if x^2<2 and \delta=\frac{(2-x^2)}{6}, show that y^2<2 whenever |x-y| < \delta

Any help would be greatly appreciated!

 

[quasar987]

How about |x-y|<(2-x^2)/6<1/3, thus y<1/3-x<1/3+sqrt(2)<2 since sqrt(2)=1.41.. and 1/3=0.33..

 

[mlearner]

That doesn't seem to be quite correct...
It needs to be proven that y^2 &lt; 2, not y &lt; 2 - the latter doesn't imply the former.
Also, I have the feeling the author expects a solution that doesn't explicitly using the value of \sqrt{2}, since no such rational number x exists such that x^2=2.

 

[uart]

The best way to approach it is to find the region in the x,y plane that satisfies the inequalies.

With the |x-y| term you're best off to consider the two cases (x>y and x<y) seperately.

For example.

Case1. Assume y>x

y < 1/3 + x - x^2 /6.

That is, we are looking for the region of the x,y plane where both y>x and y < 1/3 + x - x^2 /6 are satisfied. It's pretty straight forward to find, just the region between a parabola and a straight line.

Repeat for the other case (y<x) and you'll soon know everything that you could wist to kinow about the solution region and the inequality y^2<2 will be immediately apparent.

 

[mlearner]

Thanks for the replies... I posted the same problem on Usenet sci.math, and was offered a simpler solution (although I managed to solve the problem myself in pretty much the same way immediately after posting the question on sci.math - funny that).
|x-y| < d (d is delta)
=> |y| < |x| + d
=> y^2 < x^2 + d^2 + 2|x|d < x^2 + d^2 + 4d
and then showing d^2 + 4d < (2-x^2)
from which the result follows.

 

[MadScientist 1000]

Will this way work?:

|x-y| < (2-x^s)/2

Substitute sqrt (2) for x, and you get

sqrt (2) - y < 0

- y < - sqrt (2)

y > sqrt (2)

y^2 > 2

_____________________________________

sqrt (2) - y > 0

- y > - sqrt (2)

y < sqrt (2)

y^2 < 2So therefore, y^2 does not equal 2, and you get your answer of y^2 < 2

 

[LukeD]

Ah no, you have to show it for all values of x and y that satisfy the inequalities, not just a specific set.

 

[mlearner]

Will this way work?:

|x-y| < (2-x^s)/2

Substitute sqrt (2) for x, and you get

sqrt (2) - y < 0

- y < - sqrt (2)

y > sqrt (2)

y^2 > 2

Nah, that's not right, for at least two reasons: (1) It is clearly specified that x^2 < 2, i.e. -sqrt(2) < x < sqrt(2), so a substitution of x = sqrt(2) is not justified, and in fact leads to |sqrt(2)-y|<0 , which can never be true (because the absolute value of a number is either greater than or equal to zero). (2) A second reason is that we're dealing in rational numbers (as specified in the original problem): sqrt(2), for us, "doesn't exist", so such a substitution would be "cheating" (even if problem (1) didn't exist).

IMO, the easiest-to-follow solution is the one outlined in my previous post. I'll "latexify" it and fill in the missing steps in my next post, in case someone's interested.