I am having some trouble understanding Nakajima's proof of the Kempf-Ness theorem in [1]. At the end (proof of Proposition 3.9(6)), his argument is basically the following:

Let $G=K_{\Bbb C}$ be a complex reductive group acting linearly on a complex vector space $V$. Let $v\in V$. If there is $X\in\operatorname{Lie}(K)$ such that the function $t\mapsto\|\exp(itX)\cdot v\|^2$ diverges, then the orbit $G\cdot v$ is closed.

Why is that? I know that if there exists $w\in\overline{G\cdot v}-G\cdot v$ then there is a one-parameter subgroup $\lambda:\Bbb C^*\to G$ such that $\lambda(t)\cdot v$ converges to $w$. But does that imply that $\exp(itX)\cdot v$ converges for all $X\in\operatorname{Lie}(K)$?

More precisely, his proof is: $$\begin{align} &\text{The function }p_v:G\to\Bbb R,g\mapsto\|g\cdot v\|^2\text{ has a minimum at }g=e\\ \implies&\exists X\in\operatorname{Lie}(K_v)^\perp\text{ such that }p_v(\exp(itX))\text{ diverges}\\ \implies& G\cdot v\text{ is closed}. \end{align}$$ The first implication is fully explained and I understand it. But the second implication is mentioned without any justification.

*Reference.*

[1] Nakajima, H. *Lectures on Hilbert schemes of points on surfaces,* Vol. 18. Providence, RI: American Mathematical Society, 1999.