Real numbers, infinite sequences

Next, we have infinite-dimensional spaces. The simplest examples are the set of infinite sequences of real numbers, TV°, and similarly C°° (but in a sense, they are actually the same space). We also have spaces which, although not identical to 7L°°, can be continuously and invertibly mapped onto TV°, at least in some neighborhood around any given point. 

If I may, I would like to advert for a moment to the recent development of non-standard analysis and sketch how infinite and infinitesimal numbers can be presented. In this I follow a beautiful expository article of Ingleton (22) though, in my haste scarcely doing him, or Luxemburg on whom he leans, full justice. Consider all infinite sequences of real numbers X (x, x, ,xn, ) and let two such entities be equivalent if they differ only in a finite number of elements i.e., X E X if x =xf for all but a finite number of n. From now on we can consider the entity X to be the equivalence class and representable by any of its members just as the rational 1/2 is the class (1/2, 2/4, 3/6,..). We... 

Since our grammatical rule does not impose any restrictions on the length of a symbol sequence, we are obviously allowed to write down infinitely long sequences. There are two kinds periodic and aperiodic sequences. With the help of the mapping A —> 0, B —> 1 and C —> 2 the symbol sequences can be interpreted as real numbers in base-3 notation. On the basis of this analogy, we call finite (or infinite periodic) sequences rational and infinite aperiodic sequences irrational . Thus, the itinerary of the trajectory shown in Fig. 2.11(a) is rational. An ex-... 

If an infinite sequence approaches a definite number, then the sequence is convergent. Thus the real sequence represented above is a convergent sequence that converges to zero. 

If you were to jump one-half the distance, then one-quarter the distance, then one-eighth the distance, and so on, will you reach the door Not in a finite number of jumps In fact, if you kept jumping forever at a rate of 1 jump per second until you are out the door, you will jump forever. Mathematically one can represent this limit of an infinite sequence of actions as the sum of the series (1/2 -i- 1/4 + 1/8 -i-...). The modem tendency is to resolve Zeno s paradox by insisting that the sum of this infinite series 1/2 -F 1/4 -F 1/8. .. is equal to 1. Because each step is done in half as much time, the actual time to complete the infinite series is no different from the real time required to leave the room. (One easy way to compute the sum of this series is to use the formula y = 1 -1"", which gives the sum of the first n terms. For n = 10, y = 0.99902.)... 

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