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?

  • 2
    $\begingroup$ If $a_n=n+1/n$ then both $a_n$ and $a_n-\frac 1n$ are increasing. $\endgroup$ – lulu Mar 9 '16 at 17:25
  • $\begingroup$ How is an=n+1/n ? $\endgroup$ – user321365 Mar 9 '16 at 17:32
  • $\begingroup$ Also can someone tell me how to format. $\endgroup$ – user321365 Mar 9 '16 at 17:33
  • $\begingroup$ To disprove a proposed theorem, it suffices to exhibit a single counterexample. I defined a particular sequence $a_n$ which, I believe, passes your requirements but which does not satisfy your conclusion hence the theorem, as stated, can not be correct. $\endgroup$ – lulu Mar 9 '16 at 17:36
  • $\begingroup$ As to formatting, there is a terrific tutorial here: meta.math.stackexchange.com/questions/5020/… $\endgroup$ – lulu Mar 9 '16 at 17:37

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


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