What is the correct representation for discrete time sequence?

I am bit confused after reading Frequency-Domain Sampling and Reconstruction of Discrete-Time Signals section from Proakis where, $$x(n)=\begin{cases} x_p(n),\;\;\;\;0\leq n \leq N-1\\ \\ 0,\;\;\;\;\;\;\;\;\;\;\; \mathrm{elsewhere}\\ \end{cases}$$ It is clear $$x(n)$$ is discrete time sequence but why $$n$$ is defined as $$0 \leq n \leq N-1$$ instead of $$n=0,1,\dots,N-1$$

• Aren't those equivalents in this case? $N$ is an integer value. – jojek Feb 10 '20 at 21:51
• Yes they are. Since $n$ is integer the sequence $x(n)$ run from $0$ to $N-1$ – jomegaA Feb 10 '20 at 21:52
• $N$ can be integer, but $n$ must be integer. – jomegaA Feb 10 '20 at 21:59
• It's assumed (if not explicitly stated) that both $N$ and $n$ are integers. – MBaz Feb 10 '20 at 22:32