If true or false please explain why thank you Let xn sin n

If true or false, please explain why.... thank you.

Let x_n = sin n and y_n = cos n. Does there exist an index sequence {n_k} of positive integers such that both {x_n_k} and {y_n_k} converge?

Solution

Solution:-Given that x_n=sinn and y_n=cosn.the sequences {x_n} and {y_n }both are bounded sequences and bounded by 1.Hence by Bolzano Weirstrass theorem,every bounded sequence is convergent.

Therefore,the above sequences are convergent sequences.

Hence there exist an index sequence {n_k} of positive integers such that both {x_nk} and {y_nk} converges.

where {x_nk} and {y_nk} both are subsequences of {x_n} and {y_n} respectively.