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could you help me with this please

Let's define the relationship $\preceq$ in $\mathbb{N}^\mathbb{N}$ as $f\preceq g$ if $f(n)\leq g(n)$ for every $n\in \mathbb{N}$ Prove that it is a partial order and also show that any finite set in this order has supremum and infimum.

I have already managed to prove that it is a partial order, but the part of the supremum and infimun has cost me, what I have tried is to proceed by induction on the cardinality of the finite set, but I cannot see how to conclude with respect to step n + 1

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I don't think induction is really necessary here.

Notice that $h$ is infimum of $S\subset\mathbb N^{\mathbb N}$ if for each $n\in\mathbb N$, $h(n)=\inf S_n$ where $S_n=\{f(n)|f\in S\}$.

Any finite set in $\mathbb N$ has an infimum, which we call $\min$. Using this I think it is easy to construct the infimum sequence. Supremum follows similarly.

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