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Proof

(related to Proposition: Subset of Powers is a Submonoid)

• Let $x$ be an element of a monoid $(X,\ast)$.
• Consider the subset $T$ of powers $x^n$, $(n=0,1,2,\ldots)$.
• According to the rules of exponentiation, since $x^0\in T$ and $x^0=e$ is the identity element, we have $e\in T$.
• For $x^n\in T$ and $x^m\in T$ we have that $(x^n)\ast(x^m)=x^{n+m}\in T$. Therefore, $T$ is closed under the binary operation $\ast$.
• By definition of a substructure, $T$ is a submonoid of $X$.
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References

Bibliography

1. Lang, Serge: "Algebra - Graduate Texts in Mathematics", Springer, 2002, 3rd Edition