[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-573079-105":53,"doc-detail-573079-en":120},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":113,"head_meta":115,"extra_data":117,"updated_unix":119},105,"en","math-149s-analysis-cheat-sheet-1-definitions","Math 149s - Analysis Cheat Sheet - 1-Definitions","","Math 149s: Analysis Cheat Sheet compiles foundational definitions and core theorems for real analysis. It covers least upper bound and greatest lower bound, limit superior/limit inferior for sequences, and equivalent formulations of limits and continuity. It introduces continuity, differentiability via the difference quotient, and set properties such as open, closed, bounded, and compact. It then summarizes key results for sequences, series, functional limits, and continuous and differentiable functions, including major convergence tests and classical theorems.",{"@graph":63,"@context":112},[64,80,95],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":36,"@type":70,"position":76},"https://docshare.wps.com/template/forms/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/math-149s-analysis-cheat-sheet-1-definitions/573079/",4,{"url":78,"name":59,"@type":81,"author":82,"headline":59,"publisher":85,"fileFormat":88,"inLanguage":57,"description":61,"dateModified":89,"datePublished":89,"encodingFormat":88,"isAccessibleForFree":90,"interactionStatistic":91},"DigitalDocument",{"name":83,"@type":84},"Ophelia","Person",{"url":68,"name":86,"@type":87},"DocShare","Organization","application/pdf","2026-10-13",true,{"@type":92,"interactionType":93,"userInteractionCount":4},"InteractionCounter",{"@type":94},"ViewAction",{"@type":96,"mainEntity":97},"FAQPage",[98,104,108],{"name":99,"@type":100,"acceptedAnswer":101},"How are lim sup and lim inf defined for sequences?","Question",{"text":102,"@type":103},"The sheet defines lim sup and lim inf using infimum and supremum over tail subsequences. Specifically, lim sup a_n is expressed as an inf_k of sup_{n≥k} a_n, and lim inf a_n as a sup_k of inf_{n≥k} a_n.","Answer",{"name":105,"@type":100,"acceptedAnswer":106},"When does a sequence limit exist using lim sup and lim inf?",{"text":107,"@type":103},"A sequence limit lim a_n exists when lim inf and lim sup both exist and share the same value. In that case, lim a_n equals both lim inf a_n and lim sup a_n.",{"name":109,"@type":100,"acceptedAnswer":110},"Which criteria determine convergence of infinite series?",{"text":111,"@type":103},"The sheet lists several: comparison test, absolute convergence/normal convergence, geometric and p-series criteria, and ratio/root tests plus the integral test. It also includes the alternating series test and notes that the L=1 cases for ratio and root tests are ambiguous.","https://schema.org",{"og:url":78,"og:type":114,"og:title":59,"og:site_name":86,"og:description":61},"article",{"robots":116,"canonical":78},"index,follow",{"doc_id":118,"site_id":56},573079,1791931872,{"code":4,"msg":5,"data":121},{"doc_id":118,"user_id":122,"nickname":83,"user_avatar":123,"doc_module":9,"category_id":35,"category_name":36,"doc_title":59,"doc_description":61,"doc_content":124,"file_id":125,"file_url":126,"file_type":127,"file_size":128,"view_count":4,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":79,"language":129,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":130,"faqs":131,"seo_title":132,"seo_description":61,"update_tm":119,"read_time":73},7971461741311,"https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826","Math 149s: Analysis Cheat Sheet  \nMatthew Rognlie  \nOctober 7, 2009  \n1 De􀀌nitions  \nIf S is some set of real numbers:  \n1. sup S is the least upper bound of S.  \n2. inf S is the greatest lower bound of S.  \nS may or may not contain its sup or inf; if it does, we say that the sup is its maximum and the inf is its minimum.  \nFor a sequence fang, we also de􀀌ne:  \n1. lim sup an = infk supnfangk  \n2. lim inf an = sup k infnfangk  \nWe can de􀀌ne the limit lim a n of a sequence in two equivalent ways:  \n1. The limit is de􀀌ned if the lim inf and lim sup of the sequence exist and have the same value, in which case lim an = liminf an = lim sup an.  \n2. lim an = c if for any 􀀏 > 0, we can 􀀌nd some N such that for all n 􀀕 N , jan 􀀀 cj \u003C 􀀏 . lim an = 1 if for any y 2 R there is N such that for all n 􀀕 N , an > y.  \nWe say that an in􀀌nite series P bn converges if the limit of its partial sums limk!1 Pkn=1 bn converges as a sequence.  \nThere are also two equivalent notions of the limit of a function f (x) as x ! y:  \n1. limx!y f(x) = c if for all sequences xn ! y , f (xn) ! c.  \n2. limx!y f(x) = c if for every 􀀏 > 0, we can 􀀌nd some 􀀎 > 0 such that for all x such that jx 􀀀 yj \u003C 􀀎 , j f (x) 􀀀 cj \u003C 􀀏 .  \nA function f is continuous at point y if limx!y f(x) = f (y) . Using our two de􀀌nitions of limits, we can write this as:  \n1. f is continuous at y if for any sequence xn ! y , f (xn) ! f(y) .  \n2. f is continuous at y if for any 􀀏 > 0, we can 􀀌nd some 􀀎 > 0 such that for all x such that jx 􀀀 yj \u003C 􀀎 , j f (x) 􀀀 f (y)j \u003C 􀀏 .  \nA function f that is continuous at x is di􀀋erentiable at x if the limit limh!0 f~~ ~~(x+hh)􀀀f(x) exists and is 􀀌nite. If so, the limit is labeled f0 (x) .  \nA subset A 􀀚 R is:  \n1. Open if for any point x 2 A, we can 􀀌nd some 􀀎 > 0 such that the set B = fy : jy 􀀀 xj \u003C 􀀎gis a subset of A.  \n2. Closed if for any sequence xn ! x, where all xn 2 A, x 2 A as well.  \n3. Bounded if supx;y2R jx 􀀀 yj \u003C 1.  \n4. Compact if it is closed and bounded.  \nThe complement of an open set is closed, and vice versa.  \n2 Facts  \nSome facts about sequences include:  \n1. Squeeze Theorem:  \n(a) If an 􀀔 cn 􀀔 bn for all n, an ! L and bn ! L, then cn ! L as well.  \n(b) If an 􀀔 bn for all n and an ! 1, then bn ! 1 as well.  \n2. Cauchy Criterion: an ! a if and only if for any 􀀏 > 0 we can 􀀌nd some N such that for all m; n 􀀕 N , jam 􀀀 an j \u003C 􀀏 .  \n3. Weierstrass Theorem: A monotonic bounded sequence converges.  \n4. Sequential Compactness: A compact subset of the reals is also sequentially compact, meaning that any sequence in it contains a convergent subsequence.  \n5. Cezaro-Stolz Theorem: Let fx ng and fy ng be two sequences of real numbers, where theyn are positive, strictly increasing, and unbounded. If limn!1 x~~ n~~y~~ ~~n~~1~~1􀀀x~~ n~~y~~ ~~n = L then lim ~~x n~~y~~ ~~n exists and is equal to L.  \n6. Cantor's Nested Intervals Theorem: If I1 􀀛 I2 􀀛 : : : is a decreasing sequence of closed intervals with lengths converging to zero, then \\In consists of one point.  \nTwo types of series are especially important:  \n1. The geometric series Pkn=0 xn has sum ~~1~~~~x~~xk . Taking k ! 1, the series converges i􀀋 jxj \u003C 1, in which case the sum if 1x .  \n2. The p-series P np converges for p > 1 (assuming p is positive) .  \nYou will often apply the comparison test, which states that if an; bn 􀀕 0, an 􀀔 bn for all n and P bn converges, then P an converges as well. If P an diverges, then so does P bn.  \nThe series P an converges absolutely if P jan j \u003C 1; absolute convergence implies normal convergence. Rearranging the terms of a convergent series is only guaranteed to leave the sum the same if the series converges absolutely. Some tests for absolute convergence (and convergence more generally) include:  \n1. Ratio Test. Letting L = limn!1 􀀌 a~~n+1~~a~~ ~~n 􀀌 , we can conclude that the series converges if L \u003C 1 and diverges if L > 1; the common case L = 1 is ambiguous.  \n2. Root Test. Letting L = limn!1 pn jan j, we can conclude that the series converges if L \u003C 1","cbCaioNPldeK5juI","https://ap.wps.com/l/cbCaioNPldeK5juI","pdf",99110,"English","# Definitions\n## Limits and sequences\n## Functions and continuity\n## Open, closed, bounded, compact\n# Facts\n## Sequence theorems\n## Series convergence tests\n## Continuous function results\n## Differentiable function tools","[{\"question\":\"How are lim sup and lim inf defined for sequences?\",\"answer\":\"The sheet defines lim sup and lim inf using infimum and supremum over tail subsequences. Specifically, lim sup a_n is expressed as an inf_k of sup_{n≥k} a_n, and lim inf a_n as a sup_k of inf_{n≥k} a_n.\"},{\"question\":\"When does a sequence limit exist using lim sup and lim inf?\",\"answer\":\"A sequence limit lim a_n exists when lim inf and lim sup both exist and share the same value. In that case, lim a_n equals both lim inf a_n and lim sup a_n.\"},{\"question\":\"Which criteria determine convergence of infinite series?\",\"answer\":\"The sheet lists several: comparison test, absolute convergence/normal convergence, geometric and p-series criteria, and ratio/root tests plus the integral test. It also includes the alternating series test and notes that the L=1 cases for ratio and root tests are ambiguous.\"}]","Math 149s - Analysis Cheat Sheet - 1-Definitions | PDF"]