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Suppose that $\liminf_{t\rightarrow\infty}\|x(t)\|\neq0$, so there exists a neighborhood $H$ of 0 and an increasing sequence $(t_n)$, tending to infinity, such that $x(t_n)\not\in H$. Let $V$ be a neighborhood that the stability suggests for $H$. By the definition, $x(t)\not\in V\cap f^{-1}(V)$, $t\geq0$. Hence, $f(x(t))\not\in V$ for $t\geq0$, whence $\liminf_{t\rightarrow\infty}f(x(t))\neq0$.