Suppose the following transversality condition:
$\lim\limits_{t \rightarrow \infty}\lambda(t)\left [x(t)-x_*(t)\right] \geq 0 \quad (1)$
This condition holds for all feasible solutions $x(t)$, while $x_*(t)$ denotes the optimal solution of an optimal control problem.
Furthermore consider now, that $\lambda(t)>0$ and $x(t)\geq 0$ for all times $t$.
Under this additional assumptions i want to show that (1) is equivalent to the following conditions:
$\lim\limits_{t \to \infty}\lambda(t)x_*(t)=0, \qquad \lim\limits_{t \to \infty}\lambda(t)\geq 0. \quad (2)$
I have shown the implication $(2) \Rightarrow (1)$, but I am struggling with the inverse implication. Any ideas?
My problem here is following: I want dot split (1) the following way:
$\lim\limits_{t \to \infty}\lambda(t)\left[x(t)-x_*(t)\right]\!=\!\lim\limits_{t \to \infty}\lambda(t)x(t)-\lim\limits_{t \to \infty}\lambda(t)x_*(t)$
But how do i know, that the terms on the righthandside exists and that these are not of the form $\infty-\infty$?