= Solution
The factorization
$$
T^n-\lambda^nI=(T-\lambda I)
\left(T^{n-1}+\lambda T^{n-2}+\cdots+\lambda^{n-1}I\right)
$$
has commuting factors because both are polynomials in $T$. If $\lambda^n\notin\sigma(T^n)$, their product is invertible. The preceding part would then make $T-\lambda I$ invertible, contradicting $\lambda\in\sigma(T)$. Therefore
$$
\boxed{\lambda\in\sigma(T)\Longrightarrow\lambda^n\in\sigma(T^n).}
$$
This proves the required inclusion without assuming the full <spectral mapping theorem>.
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