Proving on the completeness theorem of real number

[akak.ak88]

1. Homework Statement

(1) ** Show that a function f is continuous at a point c if and only
if for every sequence (xn) of points in the domain of f such that
xn ! c we have limn!1 f(xn) = f(c) = f(limn!1 xn).
(2) Let A be a non-empty subset of R. Dene A := fxjx 2 Ag.
Show the following statements.
(a) A has a supremum if and only if A has an inmum, in
which case we have inf(A) = sup A.
(b) A has an inmum if and only if A has a supremum, in
which case we have sup(A) = inf A.
(3) Show that the completeness axiom of real number system (i.e.
the Least Upper Bound Property) is equivalent to the Greatest
Lower Bound Property: Every non-empty set A of real numbers
that has a lower bound has a greatest lower bound.
HINT: Use (2).
(4) * (Monotone Property) Suppose that A B R, where A =6 ;
and B =6 ;. Show the following statements.
(a) If B has a supremum, then A has also a supremum, and
sup A sup B.
(b) If B has an inmum, then A has also an inmum, and
inf A inf B.
(5) ** Let A and B be non-empty subsets of R such that a b for
all a 2 A and b 2 B. Show that A has a supremum and B has
an inmum, and sup A inf B


2. Homework Equations
The completeness axiom



3. The Attempt at a Solution
I am seriously clueless on how to approach... but I still tried something
But my method seems a bit weird...

 

[SammyS]

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