I have 5 questions in this file, but you just need to do 4th and 5th.

but I just have one hour!

please help me quickly, thank you so much

 

 

1, Given: A is an open subset of , ∩ ≠ ∅. Show: ∩ ≠ ∅.
2.Given: U is an open subset of a) Show: ⊆ ()∘
b) Can we have ≠ ()∘ ?
3. Given: A is a subset of . Show: ′ = ()′ (A’=set of all limit
points of A)
4.For ⊆ , diameter of A, diam(A) = sup {∥ − ∥: , ∈ }
a) Show that diam(A)<∞ if and only if, A is bounded.
b) Show that if A is compact then there exist x0,y0 ∈ A such that =∥ 0 − 0 ∥
5.Given a sequence {Xn} in , ∈ , consider the statements
a) given any nbd(=neighborhood) U of x, {n: Xn ∈ } is infinite.
b) given any nbd U of x, {Xn} is frequently in U (for every n ∈ ,
there exists k>n, such that ∈ ).
c) there is a subsequence of {Xn} that converges to X.
show: 1): a⟹ 2): b⟹ 3): c⟹

Attachments:

• 1492589545-Questions.pdf