in our lecture course we discussed that for finite field extensions the following equivalence holds:
Let $K \subseteq M \subseteq L$ finite field extensions.
$L / K$ separable $\Leftrightarrow L/M$ and $M/K$ are separable
I guess for infinite field extensions $\Rightarrow$ ist also true, because the minimal polynoms divide each other.
But in $\Leftarrow$ we used the fact, that the field extensions are finite. Does this direction still hold in infinite field extensions?
Yes, in any algebraic extensions. Take $a\in L$, let $f=\sum_{n=0}^d c_n x^n\in M[x]$ be its minimal polynomial and consider $F=K(c_0,\ldots,c_n)$.
$F(a)/K$ is a finite extension and it is separable (*) because $F(a)/F$ and $F/K$ are separable. Whence $K(a)/K$ is separable and so is $L/K$.
(*) is proven by $|Hom_K(F(a),\overline{L})|= |Hom_K(F,\overline{L})|\deg(f)=[F:K][F(a):F]=[F(a):K]$.