An Elementary Proof on the Limit Superior and Limit Inferior of cos(n) and sin(n)

We present an elementary proof of lim sup n cos(n) = lim sup n sin(n) = 1, and lim inf n cos(n) = lim inf n sin(n) = -1.Our approach avoids the use of the Kronecker density theorem, equidistribution theorem, or any concepts from Diophantine approximations, providing a more direct and accessible argument.