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how to prove this set is closed

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Yes. So long as by "0 <= x <= 1" you mean the subset of R\^2, not R\^1.
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I can't remember how you prove closure of [0,1], but to prove closure of A, isn't it sufficient to show it contains its boundary?
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> would it be ok

This depends on the specific instructions you were given. Sometimes, exercises ask you not only to prove something, but to do so via a particular method.

But yes: *in principle*, proving that a given set is the intersection of arbitrarily many closed sets will establish that the given set is itself closed. In fact, it might be easier to prove *A* is the intersection of *three* sets:

- *F*\_1 := { (*x*,*y*) : 0 ≤ *x* ≤ 1 },

 *F*\_2 := { (*x*,*y*) : 0 ≤ *y* }, and

 *F*\_3 := { (*x*,*y*) : *y* ≤ *x* }.

That, however, may beg the question: how do you intend to prove that *these* sets are closed?


If your directions require you to proved closedness *via sequences*, that would be not only a viable strategy, but perhaps the only one you're allowed to pursue. Other possibilities would depend on your background. (E.g., can you use that polynomial functions are continuous? That the continuous preimage of any closed set is closed?) The more you can elaborate on the instructions you have, as well as your background up to this point, the more likely we can offer productive advice.

Anyway, I hope this helps some. Good luck!

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