Consider the gradient flow in $\mathbb{R}^n$ of a (say smooth) function $U:\mathbb{R}^n\to \mathbb{R}$ which is uniformly convex. i.e for $t\geq 0$ consider
$$x'=-\nabla U(x),~~x(0)=x_0.$$
Consider the following Euler scheme with time step $h$, which using the convexity of the norm $\|\cdot \|^2$ and $U$ we can write as a optimisation problem :
\begin{align*} x'(t)&=- \nabla U(x(t)) \\ % a particle descending down an energy potential. \frac{x^{n+1}_h-x^n_h}{h}+ \nabla U(x^{n+1}_h)&=0 \\ % backward euler approximation \nabla \Big( \frac{\|x-x^n_h\|^2}{2h} + U(x) \Big) \Big|_{x=x^{n+1}_h}&=0\\ % use definition of grad operator x^{n+1}_h &=\text{argmin}_{x\in \mathbb{R}^n} \Big\{ \frac{\|x-x^n_h\|^2}{2h} + U(x) \Big\}\\ \end{align*}
$\textbf{Now my question}$ is how can I use the uniform convexity of $U$ to see that the iterates $x^n_h$ are uniformly bounded in $n$? ( Or how can I show that there exists a subsequence $n_k$ for which $x_{n_{k}}$ are uniformly bounded.