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leehuan's All-Levels-Of-Maths SOS thread (1 Viewer)

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parad0xica

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That doesn't look correct. The corollary of the triangle inequality does not state |a-b| < |a|-|b|

the closest I know of is ||x|-|y||<|x-y|

But that contradicts what you suppose.
What about

|x + y| < |x| + |y|

Let y = -y,

|x - y| < |x| + |-y| = |x| + |y|?
 

leehuan

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Please check this proof. I feel I might've accidentally assumed too much





 

leehuan

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Redo with the triangle inequality amended to not use the corollary:







EDIT: WOAH OK WAIT..
 
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Paradoxica

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Okay, here's what I have thought of... Something similar appeared in the lecture I crashed that day... IDK

 

leehuan

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Okay, here's what I have thought of... Something similar appeared in the lecture I crashed that day... IDK

It did but that looks so tedious to deal with... nor do I actually know HOW to deal with it
 

leehuan

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Edit: Nvm if sin^-1(-x)=-sin^-1(x) then the result might fall out by just changing the last few steps.

But just remind me why we don't have to consider [-pi/2,0]?
 
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leehuan

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WLOG it is. Anyway bump

Please check this proof. I feel I might've accidentally assumed too much





+ New question (under a continuity topic, but I've done limits as well of course)



Edit nvm this question, just realised stationary points can be combined with the intermediate value theorem. Just check the proof please.
 
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parad0xica

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WLOG it is. Anyway bump



+ New question (under a continuity topic, but I've done limits as well of course)



Edit nvm this question, just realised calculus can be combined with the intermediate value theorem. Just check the proof please.
Proof looks good.
 

leehuan

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Just out of curiosity, is there a name we define for the family of polynomials f(x)=x^n (n in Z+) rather than f(x) = ax^n+bx^(n-1)+etc
 

InteGrand

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This sounds like it's asking me to prove the max-min theorem, or at least the latter. How does this work?
 

seanieg89

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This sounds like it's asking me to prove the max-min theorem, or at least the latter. How does this work?
Call it the extreme value theorem, the term "max-min" and "min-max" occur a ton in other contexts.

How you should prove it will depend on exactly what you have learned, typically you assume this theorem without proof in most first year courses. (For good reason, discussion of things like compactness distract a bit from learning calculus which is the main point of first year calculus course).

Have you heard of any of the following?

-Compactness
-Bolzano-Weierstrass theorem
-Heine Borel theorem
 
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