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Given $x_1 := a > 0$ and $x_{n+1} := x_n + \frac{1}{x_n}$ for $n \in \mathbb{N}$, determine whether $(x_n)$ converges or diverges.

Since $x_1 > 0$, it seems obvious that the sequence is strictly increasing and always positive because we are always adding a positive number to each subsequent element in the sequence.

The trickier part is to show whether $(x_n)$ is bounded or not. If $(x_n)$ is bounded, then $\exists M \in \mathbb{R}$ such that

\begin{align}|x_n| \leq M ~\forall n \in \mathbb{N}\tag{1} \end{align}

Alternatively, I think this means that $M$ could be a supremum of the sequence and another way to rewrite $(1)$ is given any $\epsilon > 0$

\begin{align} x_n + \epsilon \leq M \tag{2}\end{align}

Choose $\epsilon = \frac{1}{x_n}$. Then

\begin{align} x_{n+1} = x_n + \frac{1}{x_n} \leq M \implies x_n + \epsilon \leq M \tag{3}\end{align}

Thus I conclude that the inequality holds $\forall n \in \mathbb{N}$ and that $(x_n)$ is convergent. Thus by the Monotone Sequence Convergence Theorem, $(x_n)$ is convergent.


The part I am not sure about is my reasoning to show that $x_n$ being bounded is correct or not.

Answers