I am trying to understand a result on algebraic groups, namely that if $\sigma:G\to G$ is an isogeny of a connected linear algebraic group over an algebraically closed field, then $\sigma$ stabilizes some Borel subgroup $B$ of $G$, i.e. $\sigma(B)\subset B$. I think that this was originally proved by Steinberg in his book 'endomorphisms of algebraic groups'. In this book the result is given in Theorem 7.2 after which it is stated that the theorem follows from Lemma 7.3. In the header of Lemma 7.3 it then says: '... and let $B$ be a Borel subgroup of $G$ fixed by $\sigma$...' This does not really seem like a good way to prove Theorem 7.2. Note that the assumption that $\sigma$ stabilizes $B$ is in fact used in the proof of Lemma 7.3: '...We can choose $b\in B$ such that $i_b\sigma$ stabilizes a maximal torus $T\subset B$...'

So my question would be if there is an alternative way to prove that this well-known result is true? And am I missing something or is the proof in this book just wrong?