Relation between $t$ and ${\chi}^2$

Question

Recently I faced a problem to find the relation between $t$ and ${\chi}^2$. We were just taught about the relation between $t$ and $F$ as well as $F$ and ${\chi}^2$.So, I proceeded in this way,

We know ${t_n}^2$ follows $F(1,n)$ distribution $..........(1)$
Again, suppose $F(n_1,n_2)$ denotes $F$ distribution with $n_1$ and $n_2$ degrees of freedom. For large values of $n_2$, $n_1F$ follows ${\chi_{n_1}}^2$

So, from $(1)$, ${t_n}^2=F(1,n)$
For large values of $n$,
$1.F=F$ will follow ${\chi_1}^2$
Therefore, ${t_n}^2={\chi_1}^2$

If my procedure is correct?

Answer 1

Simply note that $t_\infty = \mathcal{N}$ and $\mathcal{N}^2 = \chi_1^2$, where $\mathcal{N}$ is the standard normal distribution.