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A sequence xn1 , xn2 , . . that corresponds to a strictly increasing sequence n1 , n2 , . . of non-zero natural numbers, which is convergent. 45 Proof. For this let x1 , x2 , . . be a bounded sequence of real numbers. Then we define S : Øx1 , x2 , . . Ù . In case that S is finite, there is a subsequence x1 , x2 , . . which is constant and hence convergent. In case that S is infinite, we choose some element xn1 of the sequence. Since S is bounded, there is a 0 such that S I1 : Ö¡aß4, aß4×.

Also, we arrive at a subsequence xn1 , xn2 , . . of x1 , x2 , . . such that xk È Ik for every k È N¦ . For given ε 0, there is k0 È N¦ such that aß2k0 ε. Further, let k, l È N¦ be such that k k0 and l k0 . Then it follows that xk È Ik0 , xl È Ik0 and therefore that xk ¡ xl aß2k0 ε . Hence xn1 , xn2 , . . 10. For the following, the Bolzano-Weierstrass theorem will be fundamental. 57. 12. Let x1 , x2 , . . , for which there is M 0 such that xn M for all n È N. Then x1 , x2 , . . is convergent.

10. For the following, the Bolzano-Weierstrass theorem will be fundamental. 57. 12. Let x1 , x2 , . . , for which there is M 0 such that xn M for all n È N. Then x1 , x2 , . . is convergent. Proof. Since x1 , x2 , . . is increasing and bounded from above, it follows that this sequence is also bounded. , a sequence xn1 , xn2 , . . that corresponds 46 to a strictly increasing sequence n1 , n2 , . . of non-zero natural numbers, which is convergent. We denote the limit of such sequence by x. Then, xn x for all n È N¦ .

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