# Homework Help: Proof l.u.b of A.

1. Oct 12, 2011

### second

1. The problem statement, all variables and given/known data
A= (4+X)/X WHEN x≥1
Find the least upper bound and greatest lower bound of the following sets.for a given ε>0,
Find a number in the set that exceeds l.u.b. A - ε and a number in the set that is smaller than
g.l.b.A + ε

2. Relevant equations

does this make sense? is it a right way to attack this question?

3. The attempt at a solution

x≥1 → 1 ≥1/x → 0<1/x≤ 1
is we multiply by 4. →0 < (1/x)4 ≤4
then add one → 1 < (4/x)+1 ≤ 5.
thus 1 is G.l.b of A AND 5 l.u.b of A.

Find a number in the set that exceeds l.u.b. A - ε ....

LET 5 is the upper bound of A. (given)
suppose K is also the upper bound of A. → 5 ≤ K
Suppose this is not the case.AND K < 5.
5-K > 0
ε = 5-K > 0
K =5 -ε SO THAT there is a number X(ε) E A thus x(ε) > 5-ε = K

2. Oct 13, 2011

### HallsofIvy

Yes, 5 is the lub (in fact, it is the maximum value) of A and 1 is the glb.

But I'm not sure what you are doing with:
Well, it's not given- you had to find that 5 is the least upper bound.

3. Oct 13, 2011

### second

would you use the same strategy for G.L.B TOO...WOULDN'T THE NEGATIVE CHANGE THE EQUALITY