Local martingale

Hello everyone ! I post this message because I have a little trouble to grasp the notion of local martingale. In particular, I do not understand the first example on this wikipedia page : http://en.wikipedia.org/wiki/Local_martingale. In fact, for sufficiently large k (where Tk=k) we have X_Tk = X, or X is not a martingale whereas X_Tk (stochastic process X stopped at Tk) is.

Thanks in advance !