We study the convergence of various sequences in noncompact connected semisimple Lie groups. Antezana, Pujals, and Stojanoff showed that the iterated Aluthge sequence in M_n(C) converges. We extend the result in a Lie groups context and show that the corresponding sequence converges. We also extend a result of Rutishauser on the convergence of the Bruhat iteration in GL_n(C) to a Lie group. Finally, we present some preliminary results on the behavior of the Bruhat sequence in a real Lie group.