How to Prove an Infinite Intersection of Closed Sets is Closed in R^n?

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To prove that an infinite intersection of closed sets in R^n is closed, one must demonstrate that the intersection contains all its limit points. If C1, C2, C3, etc., are closed sets, then the intersection c = ∩k=1∞ Ck is also closed. The approach involves considering a point z in the closure of C, where a sequence {zn} in C converges to z. By showing that z must also belong to C, the proof is established. This method effectively confirms that the infinite intersection retains the closed property.
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I need help with this question which askes to prove it. Anyone has any idea?

If \text C_{1} , C_{2} , C_{3} , … are all closed sets in \text R^n , then the set \text c = I_{k=1}^{\infty}c_{k} is also a closed set in \text R^n .
 
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trap said:
I need help with this question which askes to prove it. Anyone has any idea?

If \text C_{1} , C_{2} , C_{3} , … are all closed sets in \text R^n , then the set \text c = I_{k=1}^{\infty}c_{k} is also a closed set in \text R^n .
You want to show that the intersection contains all of its limit points. Try letting z be in cl(C), the closure of C, such that a sequence {z_n} in C converges to z. Then show that z must be in C.
 

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