If $\mathrm{char}k\nmid \mathrm{\mid }G\mathrm{\mid }$, every finite-dimensional representation of $G$ is completely reducible

If $V=0$, then [L1] makes $V$ completely reducible as the empty direct sum.

Assume now that ${\dim }_{k}V>0$, and as induction hypothesis suppose every representation of smaller dimension is completely reducible. Among the nonzero subrepresentations of $V$, choose one with least positive dimension, using [A1]. It is irreducible by [L2], since any proper nonzero subrepresentation would have smaller positive dimension. Call this irreducible subrepresentation $W$. By [L3], there is a subrepresentation $U$ with $V=W\oplus U$.

The subrepresentation $W$ is nonzero, so ${\dim }_{k}U<{\dim }_{k}V$. The induction hypothesis therefore makes $U$ completely reducible. Using [L1] to expand that decomposition and adjoining the irreducible summand $W$, one gets a direct-sum decomposition of $V$ into irreducible subrepresentations. Hence $V$ is completely reducible.