I recently asked a question about nested sigma (summation) notation and this question is to attempt to confirm that the answer I got was correct. In it, one of the answers (left by Rahul) for nesting sums was using an expression like this (if you look at Rahul's answer, this wasn't the answer he left which used sets but I decided to not use that one until I understand how that notations works):
$$E.q: (1.1)$$
$$\sum_{\substack{0\le m_1<n\\\\\vdots\\\\0\le m_{n-1}<m_{n-2}}}m_{n-1}$$
To simplify the expression which I proposed:
$$E.q: (1.2)$$
$$\sum_{m_1=0}^{n-1}\dots\sum_{m_{n-1}=0}^{m_{n-2}-1}m_{n-1}$$
In both cases, the amount of sigma signs should extend until there are $n-1$ sigma signs, where $n$ is any number chosen from the set of natural numbers ($\Bbb N$). So in the case of $n=3$, you would get:
$$E.q: (1.3)$$
$$\sum_{\substack{0\le m_1<5\\\\\vdots\\\\0\le m_{4}<m_{3}}}m_{4}=1$$
Which should extend to:
$$E.q: (1.4)$$
$$\sum_{m_1=0}^{4}\sum_{m_2=0}^{m_{1}-1}\sum_{m_3=0}^{m_{2}-1}\sum_{m_4=0}^{m_{3}-1}m_4=1$$
My question is, when I'm trying to compress multiple sigma signs into one like in $E.q: (1.1)$ and $E.q: (1.3)$, is the notation used correct?