Explain the meaning of the phrase least upper bound; state the least upper bound property of the real numbers. Use the least upper bound property to show that a bounded, increasing sequence of real numbers converges.

Suppose that $a_{n}, b_{n} \in \mathbb{R}$ and that $a_{n} \geqslant b_{n}>0$ for all $n$. If $\sum_{n=1}^{\infty} a_{n}$ converges, show that $\sum_{n=1}^{\infty} b_{n}$ converges.