I want to show that
Every real number $x$ is the supremum of a set of rational numbers $A$.
My attempt
Let $A := \{r \in \mathbb{Q} \,|\, r \lt x\}$ be a set of rational numbers which exists $\forall x \in \mathbb{R}$. $A$ is bounded from above by $x$ per definition and is non-empty (for example $\lfloor x-1 \rfloor \in A$). Thus $\sup(A)$ exists. I claim that $\sup(A) = x$ and show the two conditions of the supremum.
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$\forall a \in A: a \le x$.
This is true by definition of $A$.
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$\forall \varepsilon \gt 0 \,\exists a \in A: a \gt x-\varepsilon$.
Because $\mathbb{Q}$ is dense in $\mathbb{R}$, $\forall x,y \in \mathbb{R} \,\exists r \in \mathbb{Q}: x \lt r \lt y$. Let's choose the rational number $r \in \mathbb{Q}$, so that $x-\varepsilon \lt r \lt x$. Then $r \in A$ by definition of $A$ and $r \gt x-\varepsilon$ as needed. So 2. is true as well.
Therefore $x = \sup(A)$ is indeed the supremum of $A$.
Questions
Is the proof correct? Do you think I could write it better or simpler?
Really any feedback is appreciated!
Best Answer
Mistake 1: The reason for $A$ being non-empty is not that $x-1 \in A$ since that may not be a rational number.
Mistake 2: "I assume that $\sup(A)=x\ldots$" Actually you're claiming that $\sup(A)=x.$
Other than this the proof seems right.