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Let Bn={(x, y)|0≤x≤n and 0≤y≤n}, where n is a nonnegative integer. Find U(n=0 to infinity)Bn and ∩(n=0 to infinity)Bn.
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n = {(x, y)|0≤x and 0≤y} = B.
Indeed, for all n Bn is a subset of B, therefore,
(1.1) U(n=0 to infinity)Bn = {(x, y)|0≤x and 0≤y} is a subset of B.
Let (x,y) be any element of B, then there exists n such that 0≤max{x,y}≤n. This means that (x,y) belongs to Bn. That is, any element of B belongs to some Bn, and
(1.2) B is a subset of U(n=0 to infinity)Bn.
Joining together (1.1) and (1.2), we conclude that U(n=0 to infinity)Bn = {(x, y)|0≤x and 0≤y} = B
2) ∩(n=0 to infinity)Bn = {(0,0)} = B0
Indeed, for all n B0 is a subset of Bn, therefore,
(2.1) ∩(n=0 to infinity)Bn includes B0.
(2.2) ∩(n=0 to infinity)Bn is a subset of Bn for all n, particularly, it is a subset of B0
Joining together (2.1) and (2.2), we conclude that ∩(n=0 to infinity)Bn = B0 = {(0,0)}