Find the least upper bound (if it exists) and the greatest lower bound (if it exists) of {0.9, 0.99, 0.999, ...}

Question

Find the least upper bound (if it exists) and the greatest lower bound (if it exists) of {0.9, 0.99, 0.999, ...}

Seems obvious that has a glb of 0.9 and an lub of 1. How can I demonstrate this?

Answer 1

You should probably derive an expression for the sequence and show that the sequence is monotonically increasing. ie $a_i = 1-10^{-i}$ show that for $i \geqslant 1$ then $a_i \geqslant a_1$.

For lub if you strongly suspect the answer is 1 then you could show that for arbitrary finite $i$ then $a_i \leqslant 1$. I'd also show that for a particular value less than 1 there is still an $i$ for which $a_i$ exceeds it.

Answer 2

Compute the sum $$\frac{9}{10}+\frac{9}{10^2}+\frac{9}{10^3}+...$$

Answer 3

Let $S$ be your set.

Since $0.9\in S$, the greatest lower bound of $S$ cannot be greater than $0.9$. On the other hand, $0.9$ is smaller than or equal to any element of $S$. In other words, $0.9$ is a lower bound of $S$. Therefore, it is the greatest lower bound.

On the other hand $1$ is greater than any element of $S$ and therefore it is an upper bound. And if $l<1$ there is some $n\in\mathbb N$ such that$$0,\overbrace{99\ldots9}^{n\text{ times}}>l.$$So, $l$ is not an upper bound of $S$ and it follows that $1$ is the lowest upper bound.