If $K/k$ is a field extension and $T,S \subset K$ are algebraically independent sets over $k$. If $|S| < \infty$ and $|T| > |S| $, show that there exists $t \in T$ such that $S \cup \{ t \}$ is also algebraically independent.
I tried this one but don't have to much ideas. I tried to show that if for every $t \in T$ there's a polynomial $f$ that satisfies $f(s_1,...,s_r, t ) = 0$, and try to draw a contradiction from that, but I don't really know what to do. Any help is appreciated.
If every $t\in T$ is algebraic over $K(S)$ then the field extension $K(S)\subset K(S\cup T)$ is algebraic. Then $\operatorname{trdeg}_KK(S\cup T)=|S|$. On the other side, $$\operatorname{trdeg}_KK(S\cup T)=\operatorname{trdeg}_KK(T)+\operatorname{trdeg}_{K(T)}K(S\cup T)\ge|T|,$$ a contradiction.