How to prove that a sequence is divergent?

Question

For this question, I know that the sequence diverges, but I don't know how to prove it. Can anyone please help me out?

Determine whether the following sequence is convergent or divergent.

$\{\sqrt{n}+1\}$

$\lim_{n\to \infty} (\sqrt{n} + 1) = \infty$

Answer 1

Note that

$$\sqrt{n}+1>\sqrt{n}\to \infty$$

thus we can simply prove $\sqrt{n}\to \infty$.

Then fix $M$ and find $\bar n$ such that

$$\sqrt{\bar n}>M\implies \bar n>M^2$$

Answer 2

Let $L$ be a positive real number. Choose a positive integer $N$ such that $N>(L-1)^2$. Then for $n\geqslant N$, we have $$\sqrt n+1\geqslant \sqrt N+1>L,$$ and hence $\sqrt n+1\stackrel{n\to\infty}\longrightarrow+\infty$.