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

[trap]

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 $$\text R^n$$ .

 

[hypermorphism]

What is I ?

 

[phoenixthoth]

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.