Closed Under Function Composition

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Problem (Closed Under Function Composition)

Suppose that $H$ is a set of permutations of $X$. Suppose also that if $\alpha,\beta\in H$, then we must have $\alpha\circ \beta\in H$. Use induction to prove that for every $n\in \mathbb{N}$, if we know $\sigma_1,\sigma_2,\ldots,\sigma_n\in H$, then we must have $$\sigma_1\circ\sigma_2\circ\cdots\circ\sigma_n\in H.$$

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