{"id":2104,"date":"2020-09-12T22:16:38","date_gmt":"2020-09-12T20:16:38","guid":{"rendered":"https:\/\/blog.atlant.is\/?p=2104"},"modified":"2021-02-21T13:36:26","modified_gmt":"2021-02-21T11:36:26","slug":"suites-de-cauchy","status":"publish","type":"post","link":"https:\/\/blog.douzeb.is\/?p=2104","title":{"rendered":"Suites de Cauchy"},"content":{"rendered":"\n

Pour Flo.<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n\n\n\n\n

Cet exercice s’inspire d’une des m\u00e9thodes classiques de construction des r\u00e9els \u00e0 partir des rationels.<\/p>\n\n\n\n

Consid\u00e9rons l’ensemble des suites de Cauchy dans $\\mathbb{R}$, not\u00e9 $\\mathbf{C}(\\mathbb{R})$.<\/p>\n\n\n\n

Soit $\\equiv_\\mathbb{R}$ la relation entre suites de Cauchy d\u00e9finie comme suit:<\/p>\n\n\n\n

$$\\forall u, v \\in \\mathbf{C}(\\mathbb{R}) \\quad
u \\equiv_\\mathbb{R} v
\\;\\stackrel{\\text{def}}{\\iff}\\;
\\text{lim }u = \\text{lim }v$$<\/p>\n\n\n\n

(1) Montrer que $\\equiv_\\mathbb{R}$ est bien d\u00e9finie.<\/p>\n\n\n\n

(2) Montrer que $\\equiv_\\mathbb{R}$ est une relation d’\u00e9quivalence.<\/p>\n\n\n\n

(3) Trouver une formulation de $\\equiv_\\mathbb{R}$ qui ne fasse pas intervenir explicitement de nombre r\u00e9el (c’est \u00e0 dire par exemple qui ne fasse pas r\u00e9f\u00e9rence \u00e0 la limite des suites $u$ et $v$).<\/p>\n\n\n\n

(4) Utiliser la formulation trouv\u00e9e au (3) pour d\u00e9finir une relation d’\u00e9quivalence $\\equiv_\\mathbb{Q}$ similaire \u00e0 $\\equiv_\\mathbb{R}$, mais pour l’ensemble $\\mathbf{C}(\\mathbb{Q})$ des suites de Cauchy dans $\\mathbb{Q}$.<\/p>\n\n\n\n

(5) Construire une bijection entre l’ensemble des r\u00e9els $\\mathbb{R}$ et le quotien des suites de Cauchy $\\mathbf{C}(\\mathbb{Q})$ par la relation $\\equiv_\\mathbb{Q}$.<\/p>\n\n\n\n

$\\Box$<\/p>\n\n\n\n

\n\n\n\n

Indices<\/h3>\n\n\n\n

La question (3) est difficile. Deux mots indice:<\/p>\n\n\n\n

  1. Entrelacement<\/li>
  2. Diff\u00e9rence<\/li><\/ol>\n\n\n\n
    \n\n\n\n

    Indices++<\/h3>\n\n\n\n
    1. Si $u$ et $v$ sont deux suites de Cauchy, consid\u00e9rer la suite w d\u00e9finie comme suit: $$\\forall n\\in\\mathbb{N}, w_{2n}= u_n\\text{ et }w_{2n+1}=v_n$$<\/li>
    2. Si $u$ et $v$ sont deux suites de Cauchy \u00e0 valeurs dans $\\mathbb{Q}$, $\\mathbb{R}$ ou $\\mathbb{C}$, consid\u00e9rer la suite w d\u00e9finie comme suit: $$\\forall n\\in\\mathbb{N}, w_n=v_n-u_n$$<\/li><\/ol>\n","protected":false},"excerpt":{"rendered":"

      Pour Flo.<\/p>\n","protected":false},"author":1,"featured_media":3428,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[24],"tags":[],"ppma_author":[76],"authors":[{"term_id":76,"user_id":1,"is_guest":0,"slug":"fred","display_name":"Fred","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/ec0326d654fdf9f66e9eb42bb34a9bc4?s=96&d=mm&r=g","description":"","first_name":"","last_name":"","user_url":""}],"_links":{"self":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts\/2104"}],"collection":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2104"}],"version-history":[{"count":12,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts\/2104\/revisions"}],"predecessor-version":[{"id":3496,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts\/2104\/revisions\/3496"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/media\/3428"}],"wp:attachment":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2104"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2104"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2104"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fppma_author&post=2104"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}