Show that lim((cos pix)^{2n}) exists for all x in R ?
I figured that the limit will be 1 when x belongs to Z and 0 otherwise.
But how can i show that the limit actually exists?
I don't really know what you mean by showing.
We all know that $\displaystyle (x)^{2n}\to 1,~|x|=1~\&~(x)^{2n}\to 0,~|x|<1.$