Answer to Question #16571 in Real Analysis for steve peters

Question

assume that lim[1+2(-1)^n]Xn = 0. Prove that lim Xn exists and find it.

Answer 1

if n=2k then lim[1+2(-1)^n]Xn=lim[1+2(-1)^(2k)]X2k=lim3*X_(2k)
if n=2k+1 then
lim[1+2(-1)^(2k+1)]Xn=lim(-1)*X_(2k+1)
then for subsequences
0=lim3*X_(2k)=3lim X_(2k) and 0=lim(-1)*X_(2k+1)=-lim X_(2k+1)

so we
have lim X_(2k)=lim X_(2k+1)=0 so limit exist and equal 0