Given $\|\cdot\|$ to be a norm over space $M$ and $(x(t))\subset M$ is a sequence of points for any $t\geq0$.
I know the function $t\to \|x(t)\|$ is continuous and strictly decreasing w.r.t. $t$. Also, $\lim_{t\to\infty}\|(x(t))\|=c>0$
My question: can we prove that the existence of minimizer $$ x(t_0):=\operatorname{argmin}_{t\geq 0}\|x(t)-x_0\| $$ where $x_0\in M$.
Also, can we show that the minimizer is unique?