Triangle Inequality for Indexed Summations

Theorem
Let $\mathbb A$ be one of the standard number systems $\N,\Z,\Q,\R,\C$.

Let $a,b$ be integers.

Let $\left[{a \,.\,.\, b}\right]$ denote the integer interval between $a$ and $b$.

Let $f : \left[{a \,.\,.\, b}\right] \to \mathbb A$ be a mapping.

Let $|\cdot|$ denote the standard absolute value.

Let $\vert f \vert$ be the absolute value of $f$.

Then we have the inequality of indexed summations:
 * $\displaystyle \left\vert \sum_{i \mathop = a}^b f(i) \right\vert \leq \sum_{i \mathop = a}^b \vert f(i) \vert$

Proof
The proof goes by induction on $b$.

Basis for the Induction
Let $b < a$.

Then all indexed summations are zero.

Because $|0| \leq |0|$ by definition of the standard absolute value, the result follows.

This is our basis for the induction.

Induction Step
Let $b \geq a$.

We have:

By the Principle of Mathematical Induction, the proof is complete.

Also see

 * Triangle Inequality for Summation over Finite Set
 * Triangle Inequality for Series