{"id":177,"date":"2016-10-05T21:56:08","date_gmt":"2016-10-06T01:56:08","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=177"},"modified":"2017-08-22T17:01:24","modified_gmt":"2017-08-22T21:01:24","slug":"cauchy-sequences-as-real-numbers","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2016\/10\/05\/cauchy-sequences-as-real-numbers\/","title":{"rendered":"Cauchy Sequences As Real Numbers!?"},"content":{"rendered":"

Introduce Cantor\u2019s Construction of\u00a0\"\u00a0\mathbb{R}\"<\/b><\/p>\n

Our understanding of the set of real numbers may derive from the durations of time and lengths in space. We think of\u00a0the real line, or continuum, as being composed of an (uncountably) infinite number of points, each of which corresponds to a real number. I introduce one way to\u00a0construct the\u00a0real numbers, developed\u00a0by a German mathematician Georg Cantor, by first talking about a famous\u00a0version of Zeno\u2019s paradox in the context of a race between Achiles (a legendary Greek warrior) and the Tortoise:<\/p>\n

“Achilles gives the Tortoise a head start of, say 10 m, since he runs at 10<\/em> \"\u00a0and the tortoise\u00a0moves at only 1<\/em> \". Then by the tim\"Tortoisee Achilles has reached the point where the Tortoise started (<\/em>\" m), the slow but steady individual will have moved on 1 m to<\/em> \" m. When Achilles reaches<\/em> \", the laboring Tortoise will have moved on 0.1 m (to<\/em> \" m). When Achilles reaches<\/em> \", the Tortoise will still be ahead by 0.01 m, and so on. Each time Achilles reaches the point where the Tortoise was, the cunning reptile will always have moved a little way ahead.”<\/em><\/p>\n

\u00a0If we\u00a0think of the distances Achilles has to travel, first 10 m to\u00a0\", from 1 m to\u00a0\", and\u00a0from\u00a00.1 m to\u00a0\"…, we can write it as a sum of a Geometric Series:<\/p>\n

\"10\u00a0\u2026<\/p>\n

Since the distance that Achilles travels to catch the tortoise is the sum of a geometric series where \"|r|, we know that the distance is finite as the series eventually converges. So Zeno’s argument is based on the assumption that you can infinitely divide space (the race track) and time (how long it takes to run).<\/p>\n

Lots of mathematicians have disputed whether the continuum can be represented as a set of points, until Cantor\u2019s work on the construction of\u00a0\"\u00a0\mathbb{R}\" was\u00a0introduced and recognized.<\/p>\n

Before we go into Cantor\u2019s work, I want to quickly review what Abbott discusses in Chapter 2 of our textbook. We define the convergence of a sequence\u00a0\" to a real number a if (by definition)\u00a0for every\u00a0\", there exists an\u00a0\" so that whenever\u00a0\", it follows that\u00a0\". We also define a Cauchy Sequence\u00a0\" if (by definition)\u00a0for every\u00a0\", there exists an\u00a0\" so that whenever\u00a0\", it follows that \". The Cauchy Criterion then guarantees that a sequence converges if and only if it\u2019s a Cauchy sequence. Note that\u00a0a real number may have more than one rational Cauchy sequence converging to it, so here’s when\u00a0equivalence relation comes in.<\/p>\n

Cantor gave his construction of the real numbers as equivalence classes of Cauchy sequences. We let\u00a0\"\u00a0\mathcal{C}\"\u00a0be the set of all Cauchy sequences in\u00a0\". Let\u00a0\" and\u00a0\" be elements in\u00a0\"\u00a0\mathcal{C}\", then we can define an equivalence relation such that\u00a0\" if and only if\u00a0\". However, we need to prove rigorously that such an equivalence relation exists.<\/p>\n

In order to show a relation is an equivalence relation, we need to show reflexivity, symmetry, and transitivity.<\/p>\n