An increasing convergent net $(x_{\alpha})$ of real numbers is bounded by its limit.

Question

Let $(x_{\alpha})$ be an increasing net, i.e. $\alpha\leq\beta\implies x_{\alpha}\leq x_{\beta}$, in $\mathbb{R}$ that converges to $x$. Can we conclude that $x_{\alpha}\leq x$ for all $\alpha$? In fact, I think one has $x=\sup_{\alpha}x_{\alpha}$ and this would prove the claim. Thanks in advance!

Answer 1

YES. Take any $\alpha$ and let $\epsilon >0$. There exists $\gamma $ such that

$x_{\beta} <x+\epsilon$ for all $\beta \geq \gamma$ Now there exists $\tau$ such that $\tau \geq \alpha$ and $\tau \geq \gamma$. It follows that $x_{\alpha} \leq x_{\tau} <x+\epsilon$. Since $\epsilon$ is arbitrary we get $x_{\alpha} \leq x$.