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give an example of a sequence for which the set of values x1,x2...is finite and there are three different convergent sub-sequences with distinct limits Solution Consider the follownig sequence: xn = 0 if n is a multiple of 3, = 1 if n is 1 modulo 3, = -1 if n is 2 modulo 3. Clearly the sequence takes only finitely many distinct values. Furthermore the subsequences x3n = 0 for all n, x3n+1 = 1 for alln, and x3n+2 = -1 for all n, so each of these three subsequences converges to different limits..

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give an example of a sequence for which the set of values x1,x2...is finite and there are three different convergent sub-sequences with distinct limits Solution Consider the follownig sequence: xn = 0 if n is a multiple of 3, = 1 if n is 1 modulo 3, = -1 if n is 2 modulo 3. Clearly the sequence takes only finitely many distinct values. Furthermore the subsequences x3n = 0 for all n, x3n+1 = 1 for alln, and x3n+2 = -1 for all n, so each of these three subsequences converges to different limits.

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  • 1. give an example of a sequence for which the set of values x1,x2...is finite and there are three different convergent sub-sequences with distinct limits Solution Consider the follownig sequence: xn = 0 if n is a multiple of 3, = 1 if n is 1 modulo 3, = -1 if n is 2 modulo 3. Clearly the sequence takes only finitely many distinct values. Furthermore the subsequences x3n = 0 for all n, x3n+1 = 1 for alln, and x3n+2 = -1 for all n, so each of these three subsequences converges to different limits.