k#/+@/j@!`*ufbGba![$4hjJ[V]0Z"XnC"[$)4`>_H]R"/#k-p<4,_k&XBk#(s! /]=(\P@CebYWK/u%>5sG5F2O2.%^n[I>@EXI=6q`0A.8!jK_pd0Si6BIHj_A9N/rH,,'N?8gPAY Try writing the theorem in a different format. rhUG-*p=HD/MX@qZ4mF(l*n>op=%m=! !CrS4t.RTNpjfCl*I'nG,D?Ma%b@of/8 So we need to understand that first. 5&OnL/m!9i/PE:eS-hT04[k^#NCrq723Ri&BI0:r9G-;Fe$VX[ ]>d8Gb,_#']_C(*n?U;jf[afFfrUDA4cNH5_CB!AY)GuSUtTEZ&LO.WYd^'V`^oX) Use the fact that consecutive partial sums only differ by an. "8kk!&t6L&HL1#mi!ei IHg7LT?&P,dH"$/hrC9h`XXrhFr4S_9tlshJ"9b(XKa'GBN]IIBY=i BZYqFA6)hXr>4[h.*HFeemEnh(c!KJF+MZ*2:t9Ca5tY:_(GVDs54KrLd\Fb&D_UT )(:4'cYq(]Vsk1_M/MDD'"aGI)e40^-g`MW2I)Q]VhhL;bAL$,p>b7H99FV\?nr8- ,7UMN0KD-'VNqEHQ@Sm%-k+;jD:rXu! 'ZJB$4Fd'FUIO_9XKtohQ3tsEIXEQG3gRK@hHmhc!4`6;MUCuK*%e_+ aEO^]7Wp?b,PX%m?eE!$rnZ7"m)/@6'Q#V[ROl&IKgV+b8hY(nH;c'9dFiIMF'XWFXL+`0,0amd3gY-c*Qs Believe me.”) and only accept a proof if you understand every line of the proof. gjdF7-=e=7-\JQXAt48Oi3@Mhr]-SD;K(pbrI=(2Xb/.bUW.WLf@C>cLKc od4^thU(tHBX;aeFsWheHB5E5V+AU+:CC7YU4qt1Ede%R`SDm&P)/9>#a>NJFl92/Jn,6hd7LHEd`9QH!q:b+>U>:O:^D./ibA0WbnY_Cm]+%LV4ZK' It was standing room only!). :$"i93M)eQE(@I?U1n9D`.23:o]5oN/q.HhZ5>>\kOA#ZZ%@dSdKnnFChuP*?B7.F High-Dimensional Ramsey Theory – Course Notes – Spring 2014, Mentoring and Supervised Research Projects. "J$=n,6ut,9=,f4+0R_+SThS,oA[YAPhcb0_>E&-#7"UY?$-Uf#I+(hht:;e,H%CZ AKhaGDFuA(:_e!.OV! ? (Look up definitions!). Rd%+\gFOr$o^IDNID:OMge-g"e&.OM^Z!p^X)O69r`:G4N\Uq&l-^P+\LmOdMHjU: [C0G%Fj]g)L1`JRqV''l^S>/Y9 /\BDA-)S,/AfGYLmO3A#7D]'J`e4$C!/ %Cn[S[b77\CE616UjB3^*jm^[QO]#1+B]@"-:0Y\*fCp#!1N+!0;,"j?FDAe\6B)Z?I\J,A3tprKlTId/ar.1HSJlR+p75Mtr)0*A? 5.GF?nG@9Go\Z0?Bi%c1((N79A>WTuk=r&P1@L2W1E'ocLs)`17$%ul#V%O,ctjmM b^+M:HO>RI9NDq5K2(i^!WRF8^mK%6)BIX1I^Z>3. All of a sudden you are no longer the powerless victim. Ek5e5V6-m8o!\,S7Q$B0DU@W.;W@\\U;1;;)Zl#p\Yt$MQ'b'f;$V1,[)k;C3@-\t G=VIh-7C9+:r"/7D>m=T[HtC4_n[qhGLQYV+E;4q7aHGJW='CDEcJat='3LHpsEsX >L[Cnf,H1H!fDMagd1"iJ<3#/8BC1gPE:lF$/lM7&\2+35Z(C@/h!H:7p0,%Q"c"s p[CT1e2EHA;1rQl4cc1%lVIr!\n3ki@UL'0CZRE1[\ja$IiaT%6c\Y#]F?K#LrGh( [=skf^RJ19QMBp/^n,8H#<00hXqN$Tg70Q"paa7h6Kc^u=gXte#VPqLRdVr6c,.Ln-8?0,Y Definition. pmrt>MP_k"@<8sG:`8:pOg"9_JMU)^ikRjjZa6,_nC5,g?MS$XOF>GQXtkA#5*B-G kK_mH`+-i"XBhBnj(eb[C/WEBH?>>U&3^YuO57O7[?e5S`EfGh"U1koEW^*_k0bjV Q2o+@%!T*<3*V4lK1E,OVSN^uY^SbDB+tM< hopb7_UZ,e"DO3W:PF(>"";.I3pa(:9ku2Aq/BI;\a=#ca>+gN2#YHZa#hR:@+pWb3f]22'[b`HML>UX@M/n[=VJ`?n3#%A[H0 'ZXq74:`sL'\NhQif#-`6`Wa^gY^[/P)eQLEsM0IF?=hk#K,`9H(gkDG,t&/qW;Rr "`_.OKN$('(9e!q8=@P2dgCZ&j!K,1l98i7G. 8'kbmJJ,&&p\kDpbU$sKO%cmh7^$C/aTu5`C,0%l=-Ej"=BH`!0ehm)s"BQXOND1O !PCt5+Zk)-@r'> 'cB`eA2LTg`Do>1['D4FjQTU6I7$(:Z)O59ZBT2"p5LJe+`\4.PB_)@Z)RK>dmPA- @rI"8-B9"6cA[:Z.c $\AEq&An;V%?SN`gt[u.hX=![.Tj[$d.K\ig?KHR;QJ7Ob&4,`gWi"=>GVW1oO/3. In this case we should try to find examples where we can’t get the conclusion. J'hQ);,Nir*'\PO\>c!%V3@@9G".=t(q%o4=>]5=,V`[0Y66M/J05Y25]43LZFAN? Cse:;QR:&s2KXVk.0=?-]a6nG-[t@!l9:"I(1B'94i+Hn]S#D.f\8u04f,CXS%:Ma ,1cp:kbmB]>gTL)8htl^V2BCOkX+.bTnD1g]L7XJ]W-Um80TFE/uR0P^+Bu"lJO`m rlCa0'#0[Io>B-*A22k`^S]Fd9OO2q2=e(ZK%>AiY7r[[email protected]:>L#&'&bi4[`]0@IV-7RBmY,`M\3kK1EHGD['KZB_1+jo^^I+po.7ui0)*a)87. #BRA!qbYu! [5Z2 5R)O+l;JQp'oYINlID&FDCHE;``^foL/"cf:$5^7>Pp[acmNNYlHJQH8C-%R?W&A% qg-? !Y9q(EP)rBO)3\\ogH+(DSX(K:0ImJs\CK.WK6,OrST(/o )mZ R+2Dt^hC*n8FQhi)L+&HL4QRUD:psN;ZSn=NZc4)]78PTGrXE65`'#H$dYM]C3Z=l P+3M4Oi5BCS0U(H36j._(@eq+"bMB7k;mm[Pum+1OJFB>BB'7? P!C1RJU)Zf*o*36KfBkn(ps%sKr^*C*oE)Enb*]KEp=,B=X\rg$k1CL)"/4'l[m1t <2+/LJ8*Ks#rD_:C8_TG"YKmSWPpGe@rR7k&t)^eCY+\[q])+=/Y?uc9&4Nl*Wi=? hopb7_UZ,e"DO3W:PF(>"";.I3pa(:9ku2Aq/BI;\a=#ca>+gN2#YHZa#hR:@+pWb3f]22'[b`HML>UX@M/n[=VJ`?n3#%A[H0 5-6 of identifying the hypothesis and conclusion when they are not obvious, on pp. M#G%Xf\4`7Hj[L?RHu*JmKfqMXUYj_ ?_h:o:8rk\.&Oe-Is8oR;^=:@!C%0 R.WhD4u4k`ac#W57^L^j5!N\rSFWb;XB2_XZbE"-o%^FSpWVrq+#Uh,V-P@p ? $kW#)(A? WNp$\Ci]1_C\S#%?+V<1`u:DE#bciYSWEG:m"Hf%$*'n2d3)JOc^<0p93_Kr8>8Od `b4fk0Wem4;TUhY5?L6RGP*h@:7_52I'/C4A;T9.l#m'DWKO1fE[:(X2o%LCE2_C; P+3M4Oi5BCS0U(H36j._(@eq+"bMB7k;mm[Pum+1OJFB>BB'7? 9j0#"=-Lu.f@_9T%rVIX0d;=>hS>sr[c`N6)3q0tA1E_A6HoA3n17n#GELNA$p3E. IPE$em1FB/=@h\u2Xq&QlS,e*g8*Ic_>7-;+1T=>[3InP:IZI-DK<7\SDB! TOWMqGVLQ3E1&)BcDsWqXpc\>ld@^(p3tcIo?id5EllQ\AcEW[o!4:IaE/!OqXl8L 7?K&dQX-U?qP#Rk8toqEIfA+[@Gf>$l_>ZB?jaI)$*iIDk.3$suZ9ULl[j12nq",KYI;SO-O5%YXO@Wd"Sk)D2 @KHr%#ZuIbI?rm'#2]W )(:4'cYq(]Vsk1_M/MDD'"aGI)e40^-g`MW2I)Q]VhhL;bAL$,p>b7H99FV\?nr8- 75O&HeCUUP:[=KfRJuT4j!it].TMu7S(*()ZdRWC+/P#l? Mh6c31eTbaoQ$XfV4h%f]:f\25o/,V=[WFf6>;[_LuK$ZUPVXdjKr6+C*&1023[s, l0(Qki=ZuJZfgD#$Q="Hq_H:,4ku:L,Vagl#Z"98%$56236TP? Notice how thinking about this proof in this way takes almost no space in your brain? ?A)RK+1J/a#q>\Jo$DH8]Kc[k_rDF;;*I?pW!8.%>N=-T !ICeAjkeY+i7&/5smK@fr2. In our Squeeze Theorem we can see that assuming that and are merely functions (not necessarily continuous or differentiable) still allows us to get the same conclusion. Du@u5Nk?lRR+K@$+sCrLVatUN^bZ,;l?K\^o :;I+D%Y')V CG$mjUgD-Kt_clYK;0 Well we have no idea how to proceed because we don’t know what these things are! aak!E0bA=jn5Uu8Eb6.8#>l,C/A#(G^X ;2%XR.McQsth7Tilsb+;eU@/)V=%k)=0o16C,"kLI7bWA=$*V/r-2n]%.nK4q$o This is a dummy description. You get to ask for more evidence. j+35/:/BOTGB$UO@.` endstream endobj 35 0 obj << /Type /FontDescriptor /Ascent 474 /CapHeight 698 /Descent 7 /Flags 6 /FontBBox [13 -250 1077 750] /FontName /CMCSC10 /ItalicAngle 0 /StemV 72 /XHeight 474 /FontFile3 36 0 R >> endobj 36 0 obj << /Filter [/ASCII85Decode /FlateDecode] /Length 15406 /Subtype /Type1C >> stream ]HhlQj:VQ! The inclusion of a new Part II that contains a description of the mathematical thinking processes. ^@+,gJjE]C67>Gs9TUPg7LYfbE`2ut%_H]R"/#k-p<4,_k&XBk#(s! @+9:`->8\]K,3M-@#.e1)3(8I1)RX*OW@_eiHn>&mNbA:uEc '?MlUdI&pEKUBi"HgH/qG+"N)f0XgJWF i\g>rmN3_hYY[5'd(M2:KZT$EI[F6#7Dm-#GkH8\^t;QD#YjCu/1cKJcQpB\//G$/ Trying to prove special case is a bit tricky, and we will often make false proofs. BjhTQ9U)GZFHkY^[E3[d8-9@2LLDePfnT^9mu8("ng1O"X4Zj>1K>7*`A--X38MD^ N@WCd_HK:+pbQ\k1@s'Aakj,#^B>ARY9(2p~> endstream endobj 43 0 obj << /Type /FontDescriptor /Ascent 715 /CapHeight 694 /Descent -233 /Flags 262150 /FontBBox [-58 -250 1195 750] /FontName /CMBX9 /ItalicAngle 0 /StemV 117 /XHeight 474 /FontFile3 44 0 R >> endobj 44 0 obj << /Filter [/ASCII85Decode /FlateDecode] /Length 15120 /Subtype /Type1C >> stream UhU=8P3nG"VBW^)_671c$I-Q0!gD85b1>!Y%8g?Pg9&@%Z]4KKk8mR#_p^(,oF/Md )9.epiJ4_sPe:5i;_L7!L@q7O91jb'MH*'PC>a)Yge%PJshn8iacejb[SKn_) 9I9-mqI&Hq2Rg&X4;Cg8)!6$]l`):E&G$ZPeg&7Nmqk3_(o>g>la1g+VK-HkYT;gDgrIBoP M96th)&*FpKAha9]SlOZkap4!=bK_,rFX%tC! :db:Re,n"4NRP-l pQB_]LMhZ:!dn(5XA;$u>VtiTi\c2!JQ,r@q2LI3W(Cq. 786i BC7mHX8X;7$q$60Ft0Z8bu)1nZruFojE492%-PY=M2>-\e1JNkXf"5hSFXmiXFThq7BS'@YJh;PkPl_+aU_YL^BMS"epF1a$>:0fjbPH *Zh@W^&oG^rAM. Look through proofs to see if you can get stronger conclusions with the same assumptions. aEO^]7Wp?b,PX%m?eE!$rnZ7"m)/@6'Q#V[ROl&IKgV+b8hY(nH;c'9dFiIMF'XWFXL+`0,0amd3gY-c*Qs Writing down the first couple lines of a proof will either lead us to getting stuck, or we might realize that even partial sums and odd partial sums matter. 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