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N /n 0 by a subsequence, that . 0//n 0 is convergent. Let x be the limit of this sequence. t/j Ä L. t/j Ä L. n /jt 0j Ä M: Thus, for all t in Œ0; 1, the sequence . t//n 0 is bounded. 9, there exists a subsequence of n which converges uniformly to a path . 8, we have L. 1 L. n / Ä M . 12 (Existence of paths of minimal length). Let X be a proper metric space, let x and y be two points in X and suppose that there exists a rectifiable path in X joining x and y. Then there exists such a path whose length is equal to the infimum of the lengths of paths that join x and y.

5, we have L. 1 L. n / D ˛. From the definition of ˛, we also have L. / ˛, which shows that L. / D ˛. The existence of a path of minimal length joining two given points in a metric space is interesting information and it leads to non-trivial properties. In particular, such a path is necessarily injective. An injective path is usually called a Jordan path. 6). 12 will be used later on in the proof of the existence of local geodesics in homotopy classes of paths with fixed endpoints. 13. Let X be a proper metric space, let x and y be two points in X and let be a rectifiable path joining x and y.

Iii) Spheres. For n 3, let S D S n 1 be the unit sphere in Euclidean space En and let d be the metric induced on S by the metric of En . S; d / is not a length space. ˛=2/, where ˛ is the angle, with value in the interval Œ0; , formed by the two rays issuing from the origin and passing through x and y. On the other hand, the length L. / of any path W Œa; b ! S joining x and y is bounded below by the length of the smallest arc of a great circle in S (that is, a Euclidean circle of maximal diameter that is contained in the sphere) joining x and y, that is, by ˛.

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