Let $\mathbb{E_1,E_2}$ be two finite dimension Euclidean spaces. $T\in\mathcal{L}(\mathbb{E_1,E_2})$ is a linear map. Given a closed set $C\subset\mathbb{E_1}$. Let $C_\infty$ denote the asymptotic cone of $C$, i.e \begin{align} C_\infty=\{v:\exists \{x_i\}\subset C,\exists \{t_i\geq0\}\to0,t_ix_i\to v\} \end{align} If ${\rm ker}T\cap C_\infty=\{0\}$, prove that $T(C_\infty)=T(C)_\infty$.
One side is simple. Now for the side: $T(C)_\infty\subset T(C_\infty):$ If $v\in T(C)_\infty$ then there exists $\{x_i\},\{t_i\}$ such that $t_iT(x_i)=T(t_ix_i)\to v$. Now I want to show that $T^{-1}v\in C_\infty$. One theorem states that if ${\rm ker}T\cap C_\infty=\{0\}$, then $T(C)$ is closed. But I have no further ideas how to prove $\{t_ix_i\}$ converges to some element.