supremum of rational numbers true or false

Question

I'm trying to figure out if this is false or true. $\sup \{a \in \mathbb{Q} : 0 \le a <1\} = 1$

I'd say it's false because we can $a=1/2$ $\sup\{1/2\}= 1/2$

Answer 1

Not quite: The set you're considering consists of every rational number at least $0$ and strictly less than $1$. In particular, $1/2$ cannot be the supremum since it's not even an upper bound. Your set contains $3/4$, for example.

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The statement is true. It's clear that the supremum of the given set is at most $1$, since $1$ is a bound on the set. On the other hand, the set contains elements arbitrarily close to $1$, such as

$$1 - \frac{1}{n}$$

for each $n = 1, 2, 3, ...$.

Answer 2

The supremum is the least upper bound. So to decide whether $1$ is the $\sup$, ask:

1. Is $1$ an upper bound for the set?
2. Can there possibly be another upper bound which is smaller? Equivalently, given any number smaller than 1, are there any elements in the set greater than this number?