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|00000000| 30 33 30 34 01 00 01 00 | 54 e7 03 7b 7d 22 4c 69 |0304....|T..{}"Li|
|00000010| 6d 69 74 69 6e 67 20 76 | 61 6c 75 65 20 73 75 70 |miting v|alue sup|
|00000020| 65 72 69 6f 72 2c 20 6c | 69 6d 69 74 69 6e 67 20 |erior, l|imiting |
|00000030| 76 61 6c 75 65 20 69 6e | 66 65 72 69 6f 72 22 20 |value in|ferior" |
|00000040| 6e 65 77 6c 69 6e 65 20 | 6e 65 77 6c 69 6e 65 0d |newline |newline.|
|00000050| 0a 62 6f 6c 64 20 22 44 | 65 66 69 6e 69 74 69 6f |.bold "D|efinitio|
|00000060| 6e 3a 20 22 20 22 69 66 | 20 22 20 28 61 5f 6e 29 |n: " "if| " (a_n)|
|00000070| 5f 7b 6e 20 69 6e 20 73 | 65 74 4e 7d 20 22 20 20 |_{n in s|etN} "  |
|00000080| 69 73 20 61 20 73 65 71 | 75 65 6e 63 65 20 6f 66 |is a seq|uence of|
|00000090| 20 72 65 61 6c 20 6e 75 | 6d 62 65 72 73 2e 20 54 | real nu|mbers. T|
|000000a0| 68 65 6e 20 79 6f 75 20 | 64 65 66 69 6e 65 22 20 |hen you |define" |
|000000b0| 6e 65 77 6c 69 6e 65 20 | 0d 0a 6c 69 6d 73 75 70 |newline |..limsup|
|000000c0| 20 66 72 6f 6d 20 7b 6e | 20 74 6f 77 61 72 64 20 | from {n| toward |
|000000d0| 69 6e 66 69 6e 69 74 79 | 7d 20 61 5f 6e 20 3d 20 |infinity|} a_n = |
|000000e0| 6c 69 6d 20 66 72 6f 6d | 20 6e 20 74 6f 77 61 72 |lim from| n towar|
|000000f0| 64 20 69 6e 66 69 6e 69 | 74 79 20 28 22 73 75 70 |d infini|ty ("sup|
|00000100| 22 20 5c 7b 20 61 5f 6b | 20 22 3a 20 22 6b 20 3e |" \{ a_k| ": "k >|
|00000110| 3d 20 6e 5c 7d 29 20 6e | 65 77 6c 69 6e 65 0d 0a |= n\}) n|ewline..|
|00000120| 6c 69 6d 69 6e 66 20 66 | 72 6f 6d 20 7b 6e 20 74 |liminf f|rom {n t|
|00000130| 6f 77 61 72 64 20 69 6e | 66 69 6e 69 74 79 7d 20 |oward in|finity} |
|00000140| 61 5f 6e 20 3d 20 6c 69 | 6d 20 66 72 6f 6d 20 6e |a_n = li|m from n|
|00000150| 20 74 6f 77 61 72 64 20 | 69 6e 66 69 6e 69 74 79 | toward |infinity|
|00000160| 20 28 22 69 6e 66 22 20 | 5c 7b 20 61 5f 6b 20 22 | ("inf" |\{ a_k "|
|00000170| 3a 20 22 6b 20 3e 3d 20 | 6e 5c 7d 29 20 6e 65 77 |: "k >= |n\}) new|
|00000180| 6c 69 6e 65 20 6e 65 77 | 6c 69 6e 65 20 6e 65 77 |line new|line new|
|00000190| 6c 69 6e 65 0d 0a 22 57 | 65 20 73 65 65 20 74 68 |line.."W|e see th|
|000001a0| 65 20 73 65 71 75 65 6e | 63 65 20 22 20 61 5f 6e |e sequen|ce " a_n|
|000001b0| 20 3d 20 28 2d 31 29 5e | 6e 28 31 2b 20 73 69 7a | = (-1)^|n(1+ siz|
|000001c0| 65 20 2a 30 2e 35 20 7b | 31 20 6f 76 65 72 20 6e |e *0.5 {|1 over n|
|000001d0| 7d 29 20 22 2c 20 22 20 | 6e 20 3e 3d 20 31 22 2e |}) ", " |n >= 1".|
|000001e0| 20 48 65 72 65 20 69 73 | 22 20 20 6e 65 77 6c 69 | Here is|"  newli|
|000001f0| 6e 65 0d 0a 7b 7d 22 73 | 75 70 20 22 20 5c 7b 61 |ne..{}"s|up " \{a|
|00000200| 5f 6b 22 3a 20 22 20 6b | 3e 3d 6e 20 5c 7d 20 3d |_k": " k|>=n \} =|
|00000210| 20 6c 65 66 74 20 6c 62 | 72 61 63 65 20 6d 61 74 | left lb|race mat|
|00000220| 72 69 78 7b 31 20 2b 20 | 73 69 7a 65 20 2a 30 2e |rix{1 + |size *0.|
|00000230| 35 20 7b 31 20 6f 76 65 | 72 20 6e 7d 22 2c 22 23 |5 {1 ove|r n}","#|
|00000240| 22 20 69 66 20 22 6e 22 | 20 69 73 20 6f 64 64 2c |" if "n"| is odd,|
|00000250| 22 23 23 31 20 2b 20 73 | 69 7a 65 20 2a 30 2e 35 |"##1 + s|ize *0.5|
|00000260| 20 7b 31 20 6f 76 65 72 | 20 7b 6e 2b 31 7d 7d 22 | {1 over| {n+1}}"|
|00000270| 2c 22 23 22 20 69 66 20 | 22 6e 22 20 69 73 20 65 |,"#" if |"n" is e|
|00000280| 76 65 6e 2e 22 7d 20 72 | 69 67 68 74 20 6e 6f 6e |ven."} r|ight non|
|00000290| 65 20 6e 65 77 6c 69 6e | 65 0d 0a 22 54 68 75 73 |e newlin|e.."Thus|
|000002a0| 20 22 20 6c 69 6d 73 75 | 70 20 66 72 6f 6d 20 7b | " limsu|p from {|
|000002b0| 6e 20 74 6f 77 61 72 64 | 20 69 6e 66 69 6e 69 74 |n toward| infinit|
|000002c0| 79 7d 20 61 5f 6e 20 3d | 20 31 22 20 61 70 70 6c |y} a_n =| 1" appl|
|000002d0| 69 65 73 2e 22 20 6e 65 | 77 6c 69 6e 65 0d 0a 22 |ies." ne|wline.."|
|000002e0| 43 6f 72 72 65 73 70 6f | 6e 64 69 6e 67 6c 79 2c |Correspo|ndingly,|
|000002f0| 20 79 6f 75 20 68 61 76 | 65 22 20 6e 65 77 6c 69 | you hav|e" newli|
|00000300| 6e 65 0d 0a 7b 7d 22 69 | 6e 66 20 22 20 5c 7b 61 |ne..{}"i|nf " \{a|
|00000310| 5f 6b 22 3a 20 22 20 6b | 3e 3d 6e 20 5c 7d 20 3d |_k": " k|>=n \} =|
|00000320| 20 6c 65 66 74 20 6c 62 | 72 61 63 65 20 6d 61 74 | left lb|race mat|
|00000330| 72 69 78 7b 2d 28 31 20 | 2b 20 73 69 7a 65 20 2a |rix{-(1 |+ size *|
|00000340| 30 2e 35 20 7b 31 20 6f | 76 65 72 20 6e 7d 29 22 |0.5 {1 o|ver n})"|
|00000350| 2c 22 23 22 20 69 66 20 | 22 6e 22 20 69 73 20 6f |,"#" if |"n" is o|
|00000360| 64 64 2c 22 23 23 2d 28 | 31 20 2b 20 73 69 7a 65 |dd,"##-(|1 + size|
|00000370| 20 2a 30 2e 35 20 7b 31 | 20 6f 76 65 72 20 7b 6e | *0.5 {1| over {n|
|00000380| 2b 31 7d 7d 29 22 2c 22 | 23 22 20 69 66 20 22 6e |+1}})","|#" if "n|
|00000390| 22 20 69 73 20 65 76 65 | 6e 2e 22 7d 20 72 69 67 |" is eve|n."} rig|
|000003a0| 68 74 20 6e 6f 6e 65 20 | 6e 65 77 6c 69 6e 65 0d |ht none |newline.|
|000003b0| 0a 22 54 68 65 72 65 66 | 6f 72 65 2c 20 22 20 6c |."Theref|ore, " l|
|000003c0| 69 6d 69 6e 66 20 66 72 | 6f 6d 20 7b 6e 20 74 6f |iminf fr|om {n to|
|000003d0| 77 61 72 64 20 69 6e 66 | 69 6e 69 74 79 7d 20 61 |ward inf|inity} a|
|000003e0| 5f 6e 20 3d 20 2d 31 22 | 2e 22 20 6e 65 77 6c 69 |_n = -1"|." newli|
|000003f0| 6e 65 46 0c 00 64 00 64 | 00 64 00 32 00 64 00 64 |neF..d.d|.d.2.d.d|
|00000400| 00 32 00 64 00 0f 00 54 | 69 6d 65 73 20 4e 65 77 |.2.d...T|imes New|
|00000410| 20 52 6f 6d 61 6e 00 00 | 00 00 00 00 00 00 00 00 | Roman..|........|
|00000420| 00 00 02 00 00 00 09 00 | 48 65 6c 76 65 74 69 63 |........|Helvetic|
|00000430| 61 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |a.......|........|
|00000440| 00 0f 00 54 69 6d 65 73 | 20 4e 65 77 20 52 6f 6d |...Times| New Rom|
|00000450| 61 6e 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |an......|........|
|00000460| 00 00 0f 00 54 69 6d 65 | 73 20 4e 65 77 20 52 6f |....Time|s New Ro|
|00000470| 6d 61 6e 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |man.....|........|
|00000480| 00 00 00 0f 00 54 69 6d | 65 73 20 4e 65 77 20 52 |.....Tim|es New R|
|00000490| 6f 6d 61 6e 00 00 00 00 | 00 00 00 00 00 00 00 00 |oman....|........|
|000004a0| 00 00 00 00 09 00 48 65 | 6c 76 65 74 69 63 61 00 |......He|lvetica.|
|000004b0| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 07 |........|........|
|000004c0| 00 43 6f 75 72 69 65 72 | 00 00 00 00 00 00 00 00 |.Courier|........|
|000004d0| 00 00 00 00 00 00 00 00 | 0a 00 05 00 0a 00 1e 00 |........|........|
|000004e0| 1e 00 00 00 00 00 00 00 | 05 00 0a 00 0a 00 00 00 |........|........|
|000004f0| 05 00 03 00 1e 00 0a 00 | 0a 00 00 00 14 00 01 01 |........|........|
|00000500| 64 00 53 07 00 75 6e 6b | 6e 6f 77 6e 00 00 00    |d.S..unk|nown... |
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