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This may seem like a trivial question (and I think I know the answer, just wanna check if it´s correct).

We have a function $f:\Bbb R^n \rightarrow\Bbb R$ and subsests $Y \subseteq X \subseteq\Bbb R^n$ and a point $x \in Y.$ Two verdicts:

a) $f$ has its minimum in $x$ on subset $X$

b) $f$ has its minimum in $x$ on subset $Y$

Does one verdict imply the other? I think that a implies b because $\forall x \in X: x \in Y$ and therefor there couldnt exist $y \in Y = \min(f): y \notin X$. Is this correct? Are there some other implications? Thanks

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up vote 0 down vote accepted

Yes, a) implies b): we can rephrase a) as "for each $x'\in X$, $f(x')\geq f(x)$", and b) as "for each $y\in Y$, $f(y)\geq f(x)$". Assuming a) and given $y\in Y$, we have $f(y)\geq f(x)$ since $y\in X$ also, and so b) follows.

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