Least upper bound help!?

Since n>=1, then:

0 < 1/n <= 1

0 > -1/n >= -1 .... multiply by -1, reversing inequality direction

1 > 1 - 1/n >= 0 .... add 1

So 1 is an upper bound and 0 is a lower bound.

When n=1 then 1-1/n = 0, so 0 is in fact the greatest lower bound.

If there is a lesser upper bound a<1, then:

1 > a >= 1-1/n .... for all n

0 > a-1 >= -1/n .... subtract 1

0 < 1-a <= 1/n .... negate, reversing inequalities

1/(1-a) >= n .... take reciprocals (except 0) and reverse inequalities

But for any number a (except 1) 1/(1-a) is a specific real number that can't be greater than every n value, so no such upper bound a<1 is possible.

x = {1, 2, 3, ...}

The smallest value of x is x = 1. At that point 1/x will be the largest it can possibly be. And then when you subtract the result from 1, you have the smallest value possible.

1 - 1/1 = 0

That's the lower bound.

As for the upper bound, you can make x as big as you like and 1/x will be getting smaller and smaller, essentially approaching 0. When you subtract 1 - 0 you get 1 as the upper bound.

GLB = 0

LUB = 1

hhhh