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How do I show that $H$ is a subgroup of $G$?
G is the set of all functions from $\mathbb{N}$ to $\mathbb{N}$ and H is the set of all the functions which are increasing.
How do I show that H is a subgroup of G?
I have no idea where to start.
Help would be appreciated.
A:
The axioms of a group are merely specifying a set of relations that together must govern the structure of the group. So in the case of a group under composition, to show that $H$ is a subgroup of $G$ means that it satisfies the following set of relations:
$(g_1,g_2)\in H\implies g_1,g_2\in H$
$(h_1,h_2)\in H\implies h_1h_2\in H$
Of course, any function is an element
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