Analytic Functions by Stanislaw Saks

By Stanislaw Saks

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Moreover, as we notice immediately, the rational circles form a denun;ierable base (cf. § 3) of the system of neighbourhoods

E(P,Q). 2) we can extract from this sequence convergent subsequence {p,. J, and then from the sequence 1 convergent subsequence {q,. J. Let p and q be the limits of it such that of the plane (cf. ,,J a a. these two sequences. 4). t:Q, .. p,. it. )=e(P, Q). We distinguish two cases: 1° q¥:-cx:i. se are by hypothesis different from to P, cx:i. 2° q =cx:i. Beginning from a certain value n, we then have bn. =cx:i. ,, ¥:-cx:i for k=l,2 ,.. 5). Con­ sequently, for sufficiently large values of n we have 1 and therefore e(an,bn)=e(a,.

2 18 INTRODUCTION. Theory of sets. 1) where t is a real number and O�t�l. The points z1, z2 are called the end-points of the segment [zuz1]. If in this segment one of the two end-points, e. g. and the other, z2, as the terminal point of the segment, then the segment will be said -to be oriented and we shall denote it by [zuz2]; however, if there is no possibility of misunderstanding, the arrow in the above symbol will be omitted and instead of "oriented segment" we shall simply say "segment". 1), when t assumes are arbitrarily distinct points, is called a straight line.

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