i'm trying to prove that if $x(t_n) \rightarrow x^*$ and $x(s_n) \rightarrow x^{**}$ for $(t_n)$ and $(s_n)$ two divergent sequences then $x^* = x^{**}$, where $x(t)$ is the solution of the following problem: \begin{equation} \begin{cases} \dot{x}(t) = - \nabla \varphi(x(t))\\ x(0) = x_0 \end{cases} \end{equation} Please note that $\varphi:R^n\rightarrow R$ is in $C^{1,1}$ (differentiable and $\nabla \varphi$ is Lipschitz-coninuous) and is also convex.
Any tips on how to prove it? Detailed explanation are also welcome.