Original post by gagafacea1These 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; vertical-align: -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,resizable=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; vertical-align: -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=%5Cdisplaystyle+%5Clim_%7Bz+%5Crightarrow+z_0%7Dz%5E2%3Dz_0%5E2','latexCode','toolbar=no,location=no,scrollbars=yes,resizable=yes,status=no,width=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; vertical-align: -1px;" alt="\epsilon >0" title="\epsilon >0" onclick="newWindow=window.open('http://www.thestudentroom.co.uk/latexrender/latexcode.php?code=%5Cepsilon+%3E0','latexCode','toolbar=no,location=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; vertical-align: -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%5Csqrt%7B%7Cz_0%7C%5E2-%5Cepsilon%7D-%7Cz_0%7C','latexCode','toolbar=no,location=no,scrollbars=yes,resizable=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,scrollbars=yes,resizable=yes,status=no,width=460,height=320,left=200,top=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; vertical-align: -5px;" alt="0<|z-z_0|<\delta" title="0<|z-z_0|<\delta" onclick="newWindow=window.open('http://www.thestudentroom.co.uk/latexrender/latexcode.php?code=0%3C%7Cz-z_0%7C%3C%5Cdelta','latexCode','toolbar=no,location=no,scrollbars=yes,resizable=yes,status=no,width=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; vertical-align: -5px;" alt="|z^2-z_0^2|=|z-z_0||z+z_0|<\delta |z+z_0|=\delta |z-z_0+2z_0|" title="|z^2-z_0^2|=|z-z_0||z+z_0|<\delta |z+z_0|=\delta |z-z_0+2z_0|" onclick="newWindow=window.open('http://www.thestudentroom.co.uk/latexrender/latexcode.php?code=%7Cz%5E2-z_0%5E2%7C%3D%7Cz-z_0%7C%7Cz%2Bz_0%7C%3C%5Cdelta+%7Cz%2Bz_0%7C%3D%5Cdelta+%7Cz-z_0%2B2z_0%7C','latexCode','toolbar=no,location=no,scrollbars=yes,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; vertical-align: -5px;" alt="\delta|z-z_0+2z_0|<\delta(|z-z_0|+|2z_0|)<\delta(\delta+2|z_0|)" title="\delta|z-z_0+2z_0|<\delta(|z-z_0|+|2z_0|)<\delta(\delta+2|z_0|)" onclick="newWindow=window.open('http://www.thestudentroom.co.uk/latexrender/latexcode.php?code=%5Cdelta%7Cz-z_0%2B2z_0%7C%3C%5Cdelta%28%7Cz-z_0%7C%2B%7C2z_0%7C%29%3C%5Cdelta%28%5Cdelta%2B2%7Cz_0%7C%29','latexCode','toolbar=no,location=no,scrollbars=yes,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/32/32d6b5ff49249e0d6fd964929facc9db.png" width="198" height="21" style="border: 0px; margin: 0px; padding: 0px; vertical-align: -5px;" alt=" \mid z^2-z_0^2 \mid<\delta(\delta+2|z_0|)" title=" \mid z^2-z_0^2 \mid<\delta(\delta+2|z_0|)" onclick="newWindow=window.open('http://www.thestudentroom.co.uk/latexrender/latexcode.php?code=%0D%0A%5Cmid+z%5E2-z_0%5E2+%5Cmid%3C%5Cdelta%28%5Cdelta%2B2%7Cz_0%7C%29','latexCode','toolbar=no,location=no,scrollbars=yes,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/50/50c3a26bb9859da917c4fb903b51a707.png" width="448" height="25" style="border: 0px; margin: 0px; padding: 0px; vertical-align: -6px;" alt=" \mid z^2-z_0^2 \mid<(\sqrt{|z_0|^2-\epsilon}-|z_0|)(\sqrt{|z_0|^2-\epsilon}+|z_0|)=\epsilon " title=" \mid z^2-z_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%5E2-z_0%5E2+%5Cmid%3C%28%5Csqrt%7B%7Cz_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','toolbar=no,location=no,scrollbars=yes,resizable=yes,status=no,width=460,height=320,left=200,top=100');" /></a>
DONE!
So how was that?