Taken from here:
http://www.nucleares.unam.mx/~alberto/apuntes/humphreys.pdf
On page 28 of the PDF or 16 of the scan there is Lie's Theorem as a corollary to the main theorem of the segment, just above.
For a proof of Lie's Theorem, the author instructs to use induction on $V$, while applying the above theorem.
I was able to show this, using the induction hypothess on $V/W$, where $W$ is the subspace spanned by the eigenvector found in the above theorem, and by showing that $gl(V/W)$ must be solvable as well. (The flag in $V/W$ which is stabilized by $gl(V/W)$ is then brought up to$V$ using the homomorphism $\phi: gl(V) \to gl(V/W)$, with $\phi(x)(v) = \phi(x) + W$)
Can you think of another, simpler, method of induction for this corollary?