Prove divergence of recursive sequence $a_0 = 0, a_{n+1} = a_{n} +1$

Question

I have to prove $a_0 = 0, a_{n+1} = a_{n} +1 (n \ge 0)$ is a divergent sequence using the definition.

I've noticed that this sequence should also be described simply as $a_n=n$. Shall I simplify the recursive form into $a_n=n$ first or it's unnecessary? How shall I proceed from here? I've seen many examples showing how to prove a sequence with boundaries it is divergent, but I failed to apply the ideas to this problem.

Thanks in advance.

Answer 1

Suppose $a_n \rightarrow a \in \mathbb{R}$. Then, $a=\lim_{n\rightarrow \infty} a_{n}=\lim_{n\rightarrow \infty} a_{n}+1=a+1$ and hence $0=1$. Woops!

Answer 2

A convergent sequence is bounded. $(a_n) =(n) $ is not.