Maybe someone can help me out, I thought I had an idea but I don't think it is right.

I am trying to prove the above. My attempt:

I am given that the lim sup Xn is finite, which I think means it exists, which I will call L. So I have a sequence [Xn] = [x1, x2, x3,... xn, xn+1,....] and another sequence [Xm] = [xn, xn+1,...]. This is where I am stuck, I don't know where the next step should be. I have an idea that if I can show [Xm] is monotonically decreasing then the sup of [Xm] is Xn and then taking the limit of Xn gives me L, which would complete the proof, I think. But, I don't know how to show that the sequence is monotonically decreasing, or if it even is... it was just an idea. Any help/hints would be appreciated!