Infinite tower of algebraic extensions For all I know, the following fact should be true: Consider an infinite tower of extensions $L_0 \subset L_1 \subset L_2 \subset \cdots$ such that $L_{i + 1} / L_i$ is algebraic for all $i \in \mathbf{N}_0$. Then $\bigcup_{n \geq 0} L_n$ is algebraic over $L_0$. 
However, I'm not sure how one would prove this. I was thinking to proceed by induction: The base case is okay by definition. Now assume that the claim holds for $i - 1$. Any $\alpha \in L_i$ can be written as a root of a polynomial in $L_{i - 1}[X]$. Here comes the point where I don't have any idea how to proceed. Could someone help me out?
Thanks!

An element of the union is an element of one of the $L_i$, thus is in a finite chain of algebraic extensions, thus is algebraic over $L_0$.
