In chapter V of Lang's graduate algebra, there is a Lemma 1.7 saying that if $E/k$ is separable and satisfies the property that there exists a positive integer $n$ for which $\alpha \in E$ has degree $\leq n$ over $k$, then we have $i)$ $[E:k]$ is finite and $ii)$ $[E:k]\leq n$
This lemma gives me a further question, is the conclusions of the lemma are still true if one drops the hypothesis that $E$ be separable over $k$?
I think it cannot hold any longer but I cannot get the counterexamples. I think perhaps I can get an example that $[E:k]$ is finite but strictly greater than $n$, and another one could be where $[E:k]$ is infinite, but I cannot think of anyone.
Any hints and detailed explanations are highly appreciated!!!!!