= Solution
For an irreducible representation $V(\lambda)$, write
$$
\lambda=\sum_i m_i\omega_i.
$$
Form
$$
T=\bigotimes_iV(\omega_i)^{\otimes m_i}.
$$
The tensor product of highest-weight vectors is killed by all positive-root spaces and has weight $\lambda$. It therefore generates a highest-weight constituent isomorphic to $V(\lambda)$. By <Complete reducibility of semisimple Lie algebra representations>, this constituent is a subrepresentation of $T$. This proves the <generation by fundamental representations> statement.
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