Yes.
------
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?