Summation/Inequality/Examples/From 1 to Pi

Example of Summation by Inequality
Let $n = 3 \cdotp 1 4$.

Then:
 * $\displaystyle \sum_{1 \mathop \le j \mathop \le n} a_j = a_1 + a_2 + a_3$

Proof

 * $\displaystyle \sum_{1 \mathop \le j \mathop \le n} a_j$

means:
 * The sum of all $a_j$ where $j \in \Z$ and $1 \le j \le n$

from which it follows that $j \in \left\{ {1, 2, 3}\right\}$.

Hence the result.