Deduce supremum of Set A

Question

Define $$Set A=\{1-\frac 1n\ | n \in\ N\}$$
A. Deduce sup A

b. Use the quanitifier definition of supremum to prove your conjecture in part (a).


My attempt at the solution: I believe sup A is 1?

Answer 1

Indeed, the supremum of $A$ is $1$. My guess is that your quantifier definition is something like the following:

$\alpha$ is equal to $\sup(A)$ if $\alpha$ is an upper bound of $A$ and if for every $\epsilon>0$, there is an element $a \in A$ such that $\alpha - \epsilon < a$.

If this disagrees with your definition significantly, please say so.

Assuming that this is what you have as your definition, we want to show $1 = \sup (A)$. So, for an $\epsilon > 0$, we want to find an element $a = 1 - \frac 1n$ (with $n \in \Bbb N$) such that $$ 1 - \epsilon < 1 - \frac 1n $$ How can I describe a suitable choice of $n$ in terms of $\epsilon$?