{"id":616,"date":"2012-11-05T21:51:44","date_gmt":"2012-11-06T05:51:44","guid":{"rendered":"http:\/\/zackmdavis.net\/blog\/?p=616"},"modified":"2012-11-05T21:44:03","modified_gmt":"2012-11-06T05:44:03","slug":"two-views-of-the-monotone-sequence-theorem","status":"publish","type":"post","link":"http:\/\/zackmdavis.net\/blog\/2012\/11\/two-views-of-the-monotone-sequence-theorem\/","title":{"rendered":"Two Views of the Monotone Sequence Theorem"},"content":{"rendered":"<p>If a sequence of real numbers (<em>a<sub>n<\/sub><\/em>) is <em>bounded<\/em> and <em>monotone<\/em> (and I'm actually going to say <em>nondecreasing<\/em>, without loss of generality), then it <em>converges<\/em>. I'm going to tell you <em>why<\/em> and I'm going to tell you <em>twice<\/em>.<\/p>\n<p>If our sequence is bounded, the completeness of the reals ensures that it has a <em>least<\/em> upper bound, which we'll call, I don't know, <em>B<\/em>, but there have to be sequence elements arbitrarily close to (but not greater than) <em>B<\/em>, because if there weren't, then <em>B<\/em> couldn't be a <em>least<\/em> upper bound. So for whatever arbitrarily small \u03b5, there's an <em>N<\/em> such that <em>a<sub>N<\/sub><\/em> &gt; <em>B<\/em> \u2013 \u03b5, which implies that |<em>a<sub>N<\/sub><\/em> \u2013 <em>B<\/em>| &lt; \u03b5, but if the sequence is nondecreasing, we also have |<em>a<sub>n<\/sub><\/em> \u2013 <em>B<\/em>| &lt; \u03b5 for <em>n<\/em> \u2265 <em>N<\/em>, which is what I've been trying to tell you\u2014<\/p>\n<p>\u2014<em>twice<\/em>; suppose by way of contraposition that our sequence is <em>not<\/em> convergent. Then there <em>exists<\/em> an \u03b5 such that for all <em>N<\/em>, there exist <em>m<\/em> and <em>n<\/em> greater or equal to <em>N<\/em>, such that |<em>a<sub>m<\/sub><\/em> \u2013 <em>a<sub>n<\/sub><\/em>| is greater or equal to \u03b5. Suppose it's monotone, without loss of generality <em>nondecreasing<\/em>; that implies that for all <em>N<\/em>, we can find <em>n<\/em> &gt; <em>m<\/em> \u2265 <em>N<\/em> such that <em>a<sub>n<\/sub><\/em> \u2013 <em>a<sub>m<\/sub><\/em> \u2265 \u03b5. Now suppose our sequence is bounded above by some bound <em>B<\/em>. However, we can actually describe an algorithm to find sequence points greater than <em>B<\/em>, thus showing that this alleged bound is really not a bound at all. Start at <em>a<sub>1<\/sub><\/em>. We can find points later in the sequence that are separated from each other by at least \u03b5, but if we do this \u2308(<em>B<\/em> \u2013 <em>a<sub>1<\/sub><\/em>)\/\u03b5\u2309 times, then we'll have found a sequence point greater than the alleged bound.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>If a sequence of real numbers (an) is bounded and monotone (and I'm actually going to say nondecreasing, without loss of generality), then it converges. I'm going to tell you why and I'm going to tell you twice. If our &hellip; <a href=\"http:\/\/zackmdavis.net\/blog\/2012\/11\/two-views-of-the-monotone-sequence-theorem\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[7],"tags":[33],"_links":{"self":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/616"}],"collection":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/comments?post=616"}],"version-history":[{"count":6,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/616\/revisions"}],"predecessor-version":[{"id":627,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/616\/revisions\/627"}],"wp:attachment":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/media?parent=616"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/categories?post=616"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/tags?post=616"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}