Summation of Zero/Indexed Summation

Theorem

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

Let $a, b$ be integers.

Let $\closedint a b$ denote the integer interval between $a$ and $b$.

Let $f_0 : \closedint a b \to \mathbb A$ be the zero mapping.

Then the indexed summation of $0$ from $a$ to $b$ equals zero:

$\ds \sum_{i \mathop = a}^b \map {f_0} i = 0$

Proof

At least three proofs are possible:

• by induction, using Identity Element of Addition on Numbers
• using Indexed Summation of Multiple of Mapping
• using Indexed Summation of Sum of Mappings

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