Proving diverges to infinity

Question

If $a_1$ is positive integer and $a_{n+1} = a_n + \frac {1}{a_n}$ , how can I prove ${a_n}$ diverges to $\infty$ ?

Answer 1

If it converge (denote $\ell$ it's limit), then $$\ell=\ell+\frac{1}{\ell}\implies 0=1,$$ then it doesn't converge. Since $(a_n)$ is increasing (easy to show it by induction), it diverge to $+\infty $.

Answer 2

$a_{k+1} = a_k + \frac{1}{a_k} > a_k$ and hence the sequence is increasing. we have by the equality $a_{k}(a_{k+1}-a_k) = 1$ hence $\sum_{1}^{M}a_{k}(a_{k+1}-a_k) = M$. By summation of parts we also have that $\sum_{1}^{M}a_{k}(a_{k+1}-a_k) = a_M^2-a_1^2 - \sum_{1}^{M}a_{k}(a_{k+1}-a_k)$ implies $a_M^2 = a_1^2 + 2M$ hence $a_M^2$ diverges in $\infty$ as the sequence is positive hence $a_n$ also diverges to $\infty$