# Bounded monotonic sequences

Students will learn how to prove a recursive sequence converges, by proving the sequence is both bounded and monotonic. Here's a proof that the following recursive sequence converges:

\begin{align} & a_1 = \sqrt{20} \\ & a_n = \sqrt{20 + \sqrt{a_{n - 1}}} \end{align}

The value $${a_n}$$ converges to, can also be viewed as

$$\sqrt{20 + \sqrt{20 + \sqrt{20 + \ldots}}}$$

Students will also learn how to find the limit of such bounded monotonic sequences, and how to express any integer as an infinite nested radical, such as the one above. A lesson, on all this, can be seen by watching this, then this.