If $H$ is dense in $\mathbb{R}$ and $\alpha\in\mathbb{R}$ then there exist a sequence $\{\alpha_n\}$ converging to $\alpha$ and $\alpha\leq\alpha_n$

Question

If $a\in\mathbb{R}$ there exists a sequence $\{\beta_n\}\subset H$ converging to $\alpha$. Then the sequence $\alpha_n=\alpha+|\beta_n-\alpha|$ has the desired property. The problem is that I'm not sure if the sequence $\{\alpha_n\}$ is in $H$.

This is used for Theorem 4.4 of Wheeden's Measure and Integral.

Answer 1

You are correct to doubt that the sequence $\alpha_n$ lies in $H$. However, you can argue as follows. Given $\alpha\in\mathbb{R}$, the interval $(\alpha,\alpha+1)$ contains an element of $H$, call it $\alpha_1$. The interval $(\alpha,\min\{\alpha_1,\frac{1}{2}\})$ also contains an element of $H$, call it $\alpha_2$. Having defined $\alpha_1,\alpha_2,\cdots,\alpha_n$, consider the interval $(\alpha,\min\{\alpha_n,\alpha+\frac{1}{n}\})$, and choose a member $\alpha_{n+1}$ of $H$ in it. Since $|\alpha-\alpha_n|<\frac{1}{n}$, The sequence $\alpha_n$ converges to $\alpha$ from above.