These proofs are killing me! But I'm finally starting to get a grasp of them. So I just wanted to check if this proof is sane and correct since it is different from the ones usually given: all of the proofs I found use the minimum function, , one way or another; this one doesn't.
Anyways here is the limit:.
Proof:
Suppose . Let and let be any complex number, where . Then:
This part is straight forward, and it follows from .
This part uses the triangle inequality. The equality here will not hold for since it is strictly less than .
Substitution.
DONE!
So how was that?

gagafacea1
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 12112015 01:58
Last edited by gagafacea1; 12112015 at 03:37. 
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 12112015 02:30
In line 1, you assume that . This isn't necessarily true: what happens if ? You also assume that , which is never true.
You've got a few inequalities that you're claming are strict that aren't.
The whole thing badly needs some explanation and adjustment to make it clearer. 
AngryRedhead
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 12112015 02:42
(Original post by gagafacea1)
These proofs are killing me! But I'm finally starting to get a grasp of them. So I just wanted to check if this proof is sane and correct since it is different from the ones usually given: all of the proofs I found use the minimum function, <a rel="nofollow" href="javascript:void(0)"><img src="http://www.thestudentroom.co.uk/latexrender/pictures/c9/c9b09d247b505b567e9a89e678bb82ba .png" width="79" height="21" style="border: 0px; margin: 0px; padding: 0px; verticalalign: 5px;" alt="\min{(x,y)}" title="\min{(x,y)}" onclick="newWindow=window.open(' http://www.thestudentroom.co.uk/latexrender/latexcode.php?code=%5Cmin%7B%28x %2Cy%29%7D','latexCode','toolbar =no,location=no,scrollbars=yes,r esizable=yes,status=no,width=460 ,height=320,left=200,top=100');" /></a>, one way or another; this one doesn't.
Anyways here is the limit:<a rel="nofollow" href="javascript:void(0)"><img src="http://www.thestudentroom.co.uk/latexrender/pictures/bd/bddac2347e38ad96a3677589470fc512 .png" width="99" height="31" style="border: 0px; margin: 0px; padding: 0px; verticalalign: 15px;" alt="\displaystyle \lim_{z \rightarrow z_0}z^2=z_0^2" title="\displaystyle \lim_{z \rightarrow z_0}z^2=z_0^2" onclick="newWindow=window.open(' http://www.thestudentroom.co.uk/latexrender/latexcode.php?code=%5Cdisplaysty le+%5Clim_%7Bz+%5Crightarrow+z_0 %7Dz%5E2%3Dz_0%5E2','latexCode', 'toolbar=no,location=no,scrollba rs=yes,resizable=yes,status=no,w idth=460,height=320,left=200,top =100');" /></a>.
Proof:
Suppose <a rel="nofollow" href="javascript:void(0)"><img src="http://www.thestudentroom.co.uk/latexrender/pictures/93/93e50e2f00ace9894edc61ad0fe45bb3 .png" width="43" height="15" style="border: 0px; margin: 0px; padding: 0px; verticalalign: 1px;" alt="\epsilon >0" title="\epsilon >0" onclick="newWindow=window.open(' http://www.thestudentroom.co.uk/latexrender/latexcode.php?code=%5Cepsilon+%3 E0','latexCode','toolbar=no,loca tion=no,scrollbars=yes,resizable =yes,status=no,width=460,height= 320,left=200,top=100');" /></a>. Let <a rel="nofollow" href="javascript:void(0)"><img src="http://www.thestudentroom.co.uk/latexrender/pictures/bb/bb4e87881d7233dac12d878343cd727e .png" width="175" height="25" style="border: 0px; margin: 0px; padding: 0px; verticalalign: 6px;" alt="\delta=\sqrt{z_0^2\epsilon}z_0" title="\delta=\sqrt{z_0^2\epsilon}z_0" onclick="newWindow=window.open(' http://www.thestudentroom.co.uk/latexrender/latexcode.php?code=%5Cdelta%3D%5 Csqrt%7B%7Cz_0%7C%5E2%5Cepsilon%7D%7Cz_0%7C','latexCode','toolbar= no,location=no,scrollbars=yes,re sizable=yes,status=no,width=460, height=320,left=200,top=100');" /></a> and let <a rel="nofollow" href="javascript:void(0)"><img src="http://www.thestudentroom.co.uk/latexrender/pictures/9d/9d5ed8764b4f74e3b56fbe649cf91c3a .png" width="10" height="10" style="border: 0px; margin: 0px; padding: 0px;" alt="z" title="z" onclick="newWindow=window.open(' http://www.thestudentroom.co.uk/latexrender/latexcode.php?code=z','latexCode ','toolbar=no,location=no,scroll bars=yes,resizable=yes,status=no ,width=460,height=320,left=200,t op=100');" /></a> be any complex number, where <a rel="nofollow" href="javascript:void(0)"><img src="http://www.thestudentroom.co.uk/latexrender/pictures/ab/abd73f685648c305bf62adb1d0a38b42 .png" width="134" height="20" style="border: 0px; margin: 0px; padding: 0px; verticalalign: 5px;" alt="0<zz_0<\delta" title="0<zz_0<\delta" onclick="newWindow=window.open(' http://www.thestudentroom.co.uk/latexrender/latexcode.php?code=0%3C%7Czz_0%7C%3C%5Cdelta','latexCode',' toolbar=no,location=no,scrollbar s=yes,resizable=yes,status=no,wi dth=460,height=320,left=200,top= 100');" /></a>. Then:
<a rel="nofollow" href="javascript:void(0)"><img src="http://www.thestudentroom.co.uk/latexrender/pictures/52/520b54988803a430641bc9d4931c01ab .png" width="466" height="21" style="border: 0px; margin: 0px; padding: 0px; verticalalign: 5px;" alt="z^2z_0^2=zz_0z+z_0<\delta z+z_0=\delta zz_0+2z_0" title="z^2z_0^2=zz_0z+z_0<\delta z+z_0=\delta zz_0+2z_0" onclick="newWindow=window.open(' http://www.thestudentroom.co.uk/latexrender/latexcode.php?code=%7Cz%5E2z_0%5E2%7C%3D%7Czz_0%7C%7Cz%2Bz_0%7C%3C%5Cdelta+% 7Cz%2Bz_0%7C%3D%5Cdelta+%7Czz_0%2B2z_0%7C','latexCode','tool bar=no,location=no,scrollbars=ye s,resizable=yes,status=no,width= 460,height=320,left=200,top=100' );" /></a>
<a rel="nofollow" href="javascript:void(0)"><img src="http://www.thestudentroom.co.uk/latexrender/pictures/48/4813a42030fdf5acc6c6bbe624b05b15 .png" width="418" height="21" style="border: 0px; margin: 0px; padding: 0px; verticalalign: 5px;" alt="\deltazz_0+2z_0<\delta(zz_0+2z_0)<\delta(\delta+2 z_0)" title="\deltazz_0+2z_0<\delta(zz_0+2z_0)<\delta(\delta+2 z_0)" onclick="newWindow=window.open(' http://www.thestudentroom.co.uk/latexrender/latexcode.php?code=%5Cdelta%7Czz_0%2B2z_0%7C%3C%5Cdelta%28%7Czz_0%7C%2B%7C2z_0%7C%29%3C%5Cdelt a%28%5Cdelta%2B2%7Cz_0%7C%29','l atexCode','toolbar=no,location=n o,scrollbars=yes,resizable=yes,s tatus=no,width=460,height=320,le ft=200,top=100');" /></a>
<a rel="nofollow" href="javascript:void(0)"><img src="http://www.thestudentroom.co.uk/latexrender/pictures/32/32d6b5ff49249e0d6fd964929facc9db .png" width="198" height="21" style="border: 0px; margin: 0px; padding: 0px; verticalalign: 5px;" alt=" \mid z^2z_0^2 \mid<\delta(\delta+2z_0)" title=" \mid z^2z_0^2 \mid<\delta(\delta+2z_0)" onclick="newWindow=window.open(' http://www.thestudentroom.co.uk/latexrender/latexcode.php?code=%0D%0A%5Cmid+ z%5E2z_0%5E2+%5Cmid%3C%5Cdelta%28%5Cd elta%2B2%7Cz_0%7C%29','latexCode ','toolbar=no,location=no,scroll bars=yes,resizable=yes,status=no ,width=460,height=320,left=200,t op=100');" /></a>
<a rel="nofollow" href="javascript:void(0)"><img src="http://www.thestudentroom.co.uk/latexrender/pictures/50/50c3a26bb9859da917c4fb903b51a707 .png" width="448" height="25" style="border: 0px; margin: 0px; padding: 0px; verticalalign: 6px;" alt=" \mid z^2z_0^2 \mid<(\sqrt{z_0^2\epsilon}z_0)(\sqrt{z_0^2\epsilon}+z_0)=\epsilon " title=" \mid z^2z_0^2 \mid<(\sqrt{z_0^2\epsilon}z_0)(\sqrt{z_0^2\epsilon}+z_0)=\epsilon " onclick="newWindow=window.open(' http://www.thestudentroom.co.uk/latexrender/latexcode.php?code=%0D%0A%5Cmid+ z%5E2z_0%5E2+%5Cmid%3C%28%5Csqrt%7B%7 Cz_0%7C%5E2%5Cepsilon%7D%7Cz_0%7C%29%28%5Csqrt%7B%7Cz_0% 7C%5E2%5Cepsilon%7D%2B%7Cz_0%7C%29%3D% 5Cepsilon%0D%0A','latexCode','to olbar=no,location=no,scrollbars= yes,resizable=yes,status=no,widt h=460,height=320,left=200,top=10 0');" /></a>
DONE!
So how was that?
I apologise in advance for not being able to help here but I couldn't resist 
gagafacea1
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 12112015 03:03
(Original post by BlueSam3)
In line 1, you assume that . This isn't necessarily true: what happens if ? You also assume that , which is never true.
You've got a few inequalities that you're claming are strict that aren't.
The whole thing badly needs some explanation and adjustment to make it clearer.
Oh and you mean the second line, where I used the triangle inequality? You're right but that shouldn't affect it much, since is strictly bigger than . I edited the post now, thanks for pointing those out.Last edited by gagafacea1; 12112015 at 03:09. 
gagafacea1
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 12112015 16:12
Bump!

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 12112015 16:40
(Original post by gagafacea1)
Bump!
With we have
But if then:
So if
So having chosen choose and the argument above will work. 
ghostwalker
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 12112015 16:41
(Original post by gagafacea1)
Bump! 
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 12112015 17:48
(Original post by ghostwalker)
Looks fine to me, though that's not the ordering I'd expect to see in an exam. You've clearly worked out a possible function for delta initially. 
Sourestdeeds
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 12112015 17:54
What level is this? Im hoping to do a maths degree but this looks crazy! hehe

DFranklin
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 12112015 17:58
(Original post by Sourestdeeds)
What level is this? Im hoping to do a maths degree but this looks crazy! hehe 
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 12112015 17:58
(Original post by Sourestdeeds)
What level is this? Im hoping to do a maths degree but this looks crazy! hehe 
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 12112015 18:01
(Original post by DFranklin)
1st year undergrad (at a decent university, at any rate). 
gagafacea1
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 12112015 18:20
(Original post by atsruser)
I found it a bit hard to read, so only scanned it. You can simplify it anyway with something like this (I'll use rather than )
(Original post by ghostwalker)
Looks fine to me, though that's not the ordering I'd expect to see in an exam. You've clearly worked out a possible function for delta initially.Last edited by gagafacea1; 12112015 at 18:22. 
DFranklin
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 12112015 18:59
(Original post by gagafacea1)
Of course I have done some scratch work, but from what I've read in multiple places, when you present the proof, you're supposed to present it in the order I did. Though maybe not in actual exams? 
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 12112015 19:03
(Original post by DFranklin)
"Supposed" is a bit strong. Suppose you want to show f is continuous at a. There's nothing wrong with starting off with the standard "take ", and then simply the assumption . You then start to put bounds on f(x)  f(a) given , piling extra conditions on to as necessary until you can finally say triumphantly: "if satisfies all these conditions then whenever , therefore f is continuous at a". 
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 12112015 19:21
(Original post by gagafacea1)
But...but...but that's how they write it in the books? And I don't see how yours is simpler than mine to be honest. Not saying you're wrong or anything, but as I said, I didn't want to use that min function.
As for simplicity, yeah, maybe you're right. It's easier to understand an argument that you've written down yourself, so I guess that may be a draw.
And as for the min business, I clearly skimmed your post a bit too quickly, and didn't see that. Sorry. A lot of these proofs do it that way though. BTW, your penultimate line looked to me like you would need delta to be a function of epsilon which is itself a function of delta, but the clever choice of delta fixes that. The min approach avoids that in a different way.
Of course I have done some scratch work, but from what I've read in multiple places, when you present the proof, you're supposed to present it in the order I did. Though maybe not in actual exams? 
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 12112015 19:46
(Original post by atsruser)
I'm not sure what you mean by "that's how they write it in the books"  with ?
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