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bedi
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Let x_n and y_n be two convergent sequences with different limits. Show that the set {x_n : n€N} n {y_n : n€N} is finite.
Attempt: by definition, for each £>0 there exists an N such that |x_n - x|<£ and similarly |y_n - y|<£ holds for every n with n>N. Take £=(x-y)/3 and assume that x_n and y_n are equal for a while. Call N_1 the number which satisfies |x_n - x|<(x-y)/3 and call N_2 which satisfies |y_n - y|<(x-y)/3. Put N=max(N_1,N_2). So after that N, the distance between x_n and y_n is minimum (x-y)/3. Hence there are only N many elements of the set. Is this correct?
Attempt: by definition, for each £>0 there exists an N such that |x_n - x|<£ and similarly |y_n - y|<£ holds for every n with n>N. Take £=(x-y)/3 and assume that x_n and y_n are equal for a while. Call N_1 the number which satisfies |x_n - x|<(x-y)/3 and call N_2 which satisfies |y_n - y|<(x-y)/3. Put N=max(N_1,N_2). So after that N, the distance between x_n and y_n is minimum (x-y)/3. Hence there are only N many elements of the set. Is this correct?