{ localUrl: '../page/cauchy_sequence.html', arbitalUrl: 'https://arbital.com/p/cauchy_sequence', rawJsonUrl: '../raw/53b.json', likeableId: '0', likeableType: 'page', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], pageId: 'cauchy_sequence', edit: '2', editSummary: '', prevEdit: '1', currentEdit: '2', wasPublished: 'true', type: 'wiki', title: 'Cauchy sequence', clickbait: 'Infinite sequences whose terms get arbitrarily close together.', textLength: '706', alias: 'cauchy_sequence', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'JoeZeng', editCreatedAt: '2016-07-07 01:45:41', pageCreatorId: 'JoeZeng', pageCreatedAt: '2016-07-05 01:14:27', seeDomainId: '0', editDomainId: '1161', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '1', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '29', text: 'A **Cauchy sequence** is a sequence in which as the sequence progresses, all the terms get closer and closer together. It is closely related to the idea of a [-convergent_sequence].\n\n## Definition\n\nIn any [-metric_space] with a set $X$ and a distance function $d$, a sequence $(x_n)_{n=0}^\\infty$ is Cauchy if for every $\\varepsilon > 0$ there exists an $N$ such that for all $m, n > N$, we have that $d(x_m, x_n) < \\varepsilon$.\n\nIn the [4bc real numbers], the distance between two numbers is usually expressed as their difference, or $|x_m - x_n|$.\n\n## Complete metric space\n\nIn a [ complete metric space], every Cauchy sequence is convergent. In particular, the real numbers are a complete metric space.', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '1', maintainerCount: '1', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: 'null', muVoteSummary: '0', voteScaling: '0', currentUserVote: '-2', voteCount: '0', lockedVoteType: '', maxEditEver: '0', redLinkCount: '0', lockedBy: '', lockedUntil: '', nextPageId: '', prevPageId: '', usedAsMastery: 'false', proposalEditNum: '0', permissions: { edit: { has: 'false', reason: 'You don't have domain permission to edit this page' }, proposeEdit: { has: 'true', reason: '' }, delete: { has: 'false', reason: 'You don't have domain permission to delete this page' }, comment: { has: 'false', reason: 'You can't comment in this domain because you are not a member' }, proposeComment: { has: 'true', reason: '' } }, summaries: {}, creatorIds: [ 'JoeZeng' ], childIds: [], parentIds: [], commentIds: [], questionIds: [], tagIds: [ 'stub_meta_tag' ], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [], subjects: [], lenses: [], lensParentId: '', pathPages: [], learnMoreTaughtMap: {}, learnMoreCoveredMap: {}, learnMoreRequiredMap: {}, editHistory: {}, domainSubmissions: {}, answers: [], answerCount: '0', commentCount: '0', newCommentCount: '0', linkedMarkCount: '0', changeLogs: [ { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15895', pageId: 'cauchy_sequence', userId: 'JoeZeng', edit: '2', type: 'newEdit', createdAt: '2016-07-07 01:45:41', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15374', pageId: 'cauchy_sequence', userId: 'JoeZeng', edit: '0', type: 'newTag', createdAt: '2016-07-05 01:14:29', auxPageId: 'stub_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15373', pageId: 'cauchy_sequence', userId: 'JoeZeng', edit: '1', type: 'newEdit', createdAt: '2016-07-05 01:14:27', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'false', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }