Mathematical Analysis

Question

Show that if A ⊃ B then inf A ≤ inf B ≤ supB  supA. Use this to show that, if A1 ⊃ A2 ⊃ · · ·, then

inf A1 ≤ inf A2 ≤ · · · ≤ inf An ≤ · · · ≤
≤ supAn ≤ · · · ≤ supA2 ≤ supA1.
Use this to show that (x_n) is increasing, (¯x_n) is decreasing, and lim x_n ≤ lim ¯x_{n .}

Answer 1

x z

A = { }

y w

B = [ ]

x = inf A

then for all n in A, x is the largest member of A

such that x<=n

Let y = inf B

then for all m in B, y is the largest member of B

such that y<=m

Since B is a proper subset of A, y is an element of A

but x is not necessarily an element of B.

Since x = inf A, then x <= y.

so inf A = x <= y = inf B.

Let z = sup A and w = sup B.

Then for any t in A, z is the smallest member of A

such that z>=t.

Likewise, for any j in B, w is the smallest member of B

such that w>=j

since B is a proper subset of A, w is an element of A.

then z>=j

so sup A = z >= j = sup B

For any set, the sup >= inf.

Therefore inf A <= inf B <= sup B <= sup A