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GmcnlS JOtnlllg these pomts to the pmn t O. Thcsc lcngtl1s will be ca lled posit ive. e. symmetric to thc point corrcspond ing to ~he length a). ero (0)- This rcsults In a onc-to-one correspondcnce betwccn all the points of the line and the ncw symhols Fig. 2. 1 / / / / o will: u11it measure. . I t is clear that the lengths of thc Jine segmcnls sn usfy Ax10ms r. By dctinition, tbe sum a+ b and thediqcrcnc~ a-b (a> b) ar~ resp:cti~·ely 1he leng1hs of the geomctrical sum and tbe gcomctncal d Jfl'cre nce o f 1he gtven.

Fig. 4 -. A COURSE OF MATHEMATICAL ANALYSIS tire /alter is infinite. = (2) will be called essentially nondecreasb1g if for any 110 there is 11 > 110 such that aa, < a,.. There hold the following properties: .. (i) 1/ a" is an essentially nondecreasing sequence aud a,. 4. Perhaps it would be easier for the reader to understand them after he has studied the next Chapter 3. 8 is rather concise. 8. Supplement a, b, . . which can now be positive or ncgative or zero. Following the ordinary rules which we shall not enumerate bere wc can define for the new symbols the arithmetical operations and the symbol ..

V::: \, \ ~~;··. 11 ... . , .. , '"\; A COURSE OF MATHEMATICAL ANALYSIS and then x,, E (e, d) for all n > N. Thus, the definition of limit can also be stated as follows: a variable Xn has as its limit a number a if the set of points x, lying outside any neighbourhood ofthe point a is either finite o1· empty. The variable in Example (vi) has no limit at all because if we suppose that it has a limit equal to a then any neighbourhood of the point a, however small in its length, contains all the elements x, with the exception of a finite number of X 11 • But it is clear that outside any interval of Iength 1/2, irrespective of its position on the real Iine, there are an infinite number of elements x, of the sequence in question, and hence the sequence has no Iimit.

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