Rearrange the alternating harmonic series to any prescribed real sum

Let \(\mathbb N\) denote the positive integers and define \[ a_n=\frac{(-1)^n}{n}. \] Let \(\alpha\in\mathbb R\). Prove that there is a bijection \(\sigma:\mathbb N\to\mathbb N\) such that \[ \sum_{n=1}^{\infty}a_{\sigma(n)}=\alpha. \]