**This is for a course, I don't want the answer just a prod in the right direction!**

I've got a problem that states

let n be an integer such that $$n\gt0$$ $$\text{Prove: }\sum\_{k=0}^n |\cos k| \ge \frac n2$$

I'm using induction to prove this. First I showed my base case:

When $P(0)$ $\sum\_{k=0}^0 |\cos o| \ge \frac 02$

Which simplifies down into $|1|\ge 0$ which holds true. So I moved to the induction step.

Assume: $P(n)$ is true such that $\sum\_{k=0}^n |\cos k| \ge \frac n2$

WTS: P(n+1) is true such that $\sum\_{k=0}^{n+1} |\cos k| \ge \frac n2$

So I broke the summation up into two parts

$\sum\_{k=0}^{n+1} |\cos k|=\sum\_{k=0}^n |\cos k|+ \cos(n+1)$ Which I can substitute back in and get

$\sum\_{k=0}^n |\cos k|+ \cos(n+1) \ge \frac {n+1}2$

I know, well I think, that this is where I use my Induction Hypothesis $\sum\_{k=0}^n |\cos k| \ge \frac n2$

But I don't now how to show my next step.

Can I just substitue $\frac n2$ in for $\sum\_{k=0}^n |\cos k|$ and continue to solve? or do I have to show more steps in the middle? or am I completely lost? If anyone can prod me into the right direction it would be much appreciated! Thanks for your help!

***First Edit***
@Bungo

This is the way I've been doing it.
Is my logic wrong? Do I need to prove the lower bound? Or is my logic okay?

So I broke the sum down and re wrote it as $\sum\_{k=0}^n + |\cos n+1| \ge \frac {n+1}2$

And then substituted from the induction hypothesis and wrote it as $\frac n2 + |\cos n+1| \ge \frac {n+1}2$

Then I simplified and got $|\cos n+1| \ge \frac 12$

So now I'm trying to prove that statement as true. Is this okay? Or should I go back and try with the triplets as you suggested? Thanks!