How to prove that a sequence is unbounded I want to ask, how to start a proof that shows a sequence to be unbounded

Define a sequence $\{X_n\}$ by

$$X_1 = 1 ,\quad X_{n+1} = X_n + \sqrt{X_n}  \quad \text{for}\ n \geq 1$$

Prove that $\{X_n\}$ is unbounded.
First,
$x_{n+1} &gt; x_n \ge 1$
for all $n$.

Second,
$x_{n+1} \ge x_n+1
$.

Therefore,
$x_n \ge n$,
so $x_n$ is unbounded.
In this case, you can show that $\lim_{n\to\infty} a_n=\infty$, which implies that the sequence is unbounded.  In other words, one shows that:
$$\text{For any $M&gt;0$, there exists $N&gt;0$ such that $n\geq N$ implies $|a_n|&gt;M$}$$
In this case, what you want to note is that $X_1 =1$, and $\sqrt{x}\geq 1$ for any $x\geq 1$.  With $X_{n+1} := X_{n+1} + \sqrt{X_{n}}$, prove that $X_n \geq n$ for all $n$, and conclude that $X_n$ is unbounded.