Let (xn) and (yn) be two sequences. Let (zn) be the shuffled sequence:
z1=x1, z2=y1, z3=x2, z4=y2.....z(2n-1)=xn, z(2n)=yn...
Prove that (zn) converges, which implies that both (xn) and (yn) converge.
Can someone help?
Thanks.
Yes.
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I did not try to solve this problem but I am thinking:
If the sequence {z_n} converges then the odd and even subsequences converge that is,
z_1,z_3,z_5,... -----> Converges
z_2,z_4,z_6,... -----> Converges
But the odd-even subsequences are the sequences x_n and y_n. Thus, {x_n} and {y_n} both converges.
Is that not true?