# Understanding an expression involving sum notation

See the following expression. $$\sum_{i=1}^n \sum_{j \in i} f(a_i)d_j - \sum_{i=1}^n \sum_{j \in i} f(b_j) d_j$$

So in the first two sums, we pick a $i$, and then we sum over all $j$ that satisfy $j \in i$ (the notation itself means nothing: just that we pick certain $j$ that satisfy something for the chosen $i$).

In the second, we do something similar, however now the $f(x)$ term now depends on $j$.

Could someone explain to me why above is less than the following expression? $$\text{'above'} \le \sum_{i=1}^n \sum_{j \in i} |f(a_i) - f(\eta_j)| d_j$$

It just pops straight up in a proof I'm reading, and I am not sure what's going on? something with triangle inequality, I'm guessing, but how does it all add up?

• In the inequality, do you mean $b_j$ instead of $\eta_j$? Commented Mar 8, 2015 at 18:01
• Oops. Yes. $b_j$. The eta is from a different proof on the same page. Commented Mar 8, 2015 at 18:03

Yes, this is just the triangle inequality. In summation for that looks like $$\left|\sum_{k=1}^n x_n\right|\le\sum_{k=1}^n|x_n|.$$