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yh05
yh05
12:28
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@user21820 These are different definitions of limit at infinity, first from Hunter's intro real analysis book, second from Bartle's intro real analysis book, and third from baby Rudin.

Is there a standard definition of limit at infinity? Which definition should I adopt?
user21820
user21820
@yh05 Second one is quite bad. It had two mistakes: (1) There is no such thing as "limit of f as x → ∞". (2) You cannot have "K = K(ε)".
The third one is a topological version to give a uniform treatment of limits whether the point is at infinity or not, and whether the limit is infinite or not.
The first one is a pedagogically simpler version, which does not even cover the case of infinite limits, but is probably useful for beginners.
@yh05 Which one (of the first or third) you should adopt depends on which one you already understand. If you understand the first definition completely, then you can think about the third one and how it works.
yh05
yh05
The third definition includes limit of t -> x when x is a real number. According to this definition, x is not necessarily a limit point of E (we discussed this before). So the topological version includes such a case?
user21820
user21820
12:47
@yh05 Interesting. That third definition requires the neighbourhood to have non-empty intersection with the domain, so even though it allows x to be somewhere isolated outside E, it still forces the limit to 'be sensible'.
In comparison, the first definition does not want to even define limits at points far away from the domain A (hence "not bounded from above").
yh05
yh05
13:31
Which of definition 1 and 3 is more commonly used?
user21820
user21820
14:01
@yh05 At undergraduate level, most people use the first one. I don't see a big deal in the difference because people simply don't talk about limits at points far away from the domain.
yh05
yh05
I see.
 
5 hours later…
Threnody
Threnody
19:18
@user21820 Q3 attempt 
If (A or B) and (B or C) and (C or A)
	A or B
	B or C
	C or A

	If A
		If B
			If C
				B and C
				conclusion
			C => conclusion
		B => C => conclusion
	A => B => C => conclusion	(used for or-elim 1)

	If B
		If B
			If C
				B and C
				conclusion
			C => conclusion
		B => C => conclusion
	B => B => C => conclusion	(used for or-elim 1)

	B => C => conclusion	(or-elim 1: A or B | used for or-elim 2)

	If A
		If C
			C and A
			conclusion
		C => conclusion
	A => C => conclusion	(used for or-elim 2)
It's so drawn out.
Whenever I write 'conclusion' I mean the obvious or-intro to get to the conclusion
I... just realised the first half is completely useless hahahaha
B => C => conclusion
To get this line I did not need 2x 3 nested ifs at all.

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