HINT cos2x sin2x 1 should help reach the answer Let xn si

HINT: cos^2(x) + sin^2(x) = 1 should help reach the answer.

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

let xn=sin n and yn=cos n

xn^2+yn^2=sin^2 n+cos^2 n=1

if nk=2pik then xnk=sin 2pi k=o for all k. so subsequence xnk converges to 0

and ynk=cos 2pi k=1 for all k, so subsequence ynk converges to 1