Is the following statement true or false?

For every odd integer $n,|$~$\frac{n}{2}$~$|=\frac{n+1}{2}.$

To answer this question, let $n$ be any odd integer. By definition of odd, $n=2k+1$ for some integer $k,$ and so$,$ by substitution, $|$~$\frac{n}{2}$~$|=|$~$\frac{2k+1}{2}$~$|.$ Starting with this equality, express $|$~$\frac{n}{2}$~$|$ in terms of $k$ as simply as possible without using the ceiling notation. The result is

$|$~$\frac{n}{2}$~$|=|$~$+\frac{1}{2}$~$|=$

Also, when $\frac{n+1}{2}$ is expressed in terms of $k$ and is completely simplified, the result is

$\frac{n+1}{2}=$

Since the expression in $k$ for $|$~$\frac{n}{2}$~$|$ is as the expression in $k$ for $\frac{n+1}{2},$ the given statement is