Jtdylktuy

Let $Omegasubseteqmathbb R^n$ be open and $f:Omegatomathbb R^n$ a continuous vector field. Show that if for some $pinOmega$ there is a solution $gamma:]a,infty[tomathbb Omega$ of $x'=f(x)$ such that $lim_{ttoinfty}gamma(t)= p$, then $p$ is a stationary solution.

I had one idea but I doubt that it works like this: If $lim_{ttoinfty} gamma(t)=p$ and $gamma$ is continuous, can I conclude $lim_{ttoinfty}gamma'(t)=0$?

If that were the case the fact that $f$ is continuous would further yield $$f(p)=f(lim_{ttoinfty}gamma(t))=lim_{ttoinfty}f(gamma(t))=lim _{ttoinfty}gamma'(t)=0$$ making $p$ a continuous solution. If my approach is wrong though and can't be fixed, how do I prove this result?

Assume $f(p)=vne 0$, then $|f(x)-v|lefrac12|v|$ for $xin B(p,delta)$ for some $δ>0$. Now per the limit assumption, $|γ(t)-p|<δ$ for $t>T$ where $T$ depends on $δ$. Then also
$$
γ(t)-γ(T)-v,(t-T)=int_T^t(f(γ(s)-v),ds
$$
so that
$$
|γ(t)-γ(T)-v,(t-T)|le frac12|v|,(t-T)
$$
which is impossible to satisfy for all $t>T$, as
$$
|γ(t)-γ(T)-v,(t-T)|ge |v|,(t-T)-2δ
$$
grows faster than the bound.