Does $a_n$ increasing imply $a_n-\frac{1}{n}$ strictly increasing?

Question

Let $(a_n)_{n=1}^\infty$ be a sequence in $\mathbb{R}$. If $(a_n)$ is increasing prove that $(a_n-1/n)$ is strictly increasing.

How can I start off this off this question via induction?

Answer 1

$(\frac{1}{n})$ is strictly decreasing, so $(-\frac{1}{n})$ is strictly increasing. Then $a_{n+1}-\frac{1}{n+1} \geq a_{n}-\frac{1}{n+1} > a_{n}-\frac{1}{n}$.