I have the following recursive function(an example from a textbook):
$$ T(n)=\begin{cases}1&n=1\\T(\lfloor\tfrac n2\rfloor)+T(\lceil\tfrac n2\rceil)+1&n>1\end{cases} $$
A recursion is used in order to prove that $T(n) = O(n)$
The hypothesis is: for each $k$ such that $\lfloor\tfrac n2\rfloor\leq k\leq n-1$ we get: $T(k)\leq c\cdot k$
My Question: why did they take a region of $k$ rather than a single value of $k$, I mean why not say: for $k = \lfloor\tfrac n2\rfloor$ we get that $T(k)\leq c\cdot k$