Therefore, By Induction

Paroles de « Therefore, By Induction »

Intro: solo distorted sawtooth electro-house lead

Verse 1

Step one. Establish the base case.
Let n equal one, and verify by hand.
The left side reads one.
The right side reads one times two over two.
One equals one.
The base case holds.

Pre-Chorus

(Definition: P of n, a statement indexed by n)
(for every n in the natural numbers)
(Definition: the well-ordering principle)
(every non-empty subset has a least element)

Chorus

Therefore, by induction.
Therefore, by induction.
For all n.
For all n.
Therefore, by induction.

Instrumental solo

Verse 2

Step two. Assume the inductive hypothesis.
Suppose P of k holds for one fixed k.
We do not prove k.
We are permitted to assume it.
The assumption is discharged at the end.
This is the method.

Pre-Chorus

(Reference: Peano, axiom five)
(Contra-indication: do not assume P of every k)
(Reference: the principle of strong induction)
(permitted only where stated)

Break

Add k plus one to both sides.
Factor. Collect the terms.
The expression becomes
k plus one, times k plus two, over two.
Which is the statement P of k plus one.
The implication holds.

Chorus

Therefore, by induction.
Therefore, by induction.
For all n.
For all n.
Therefore, by induction.

Verse 3

Step three. The inductive step is closed.
P of one is true.
P of k implies P of k plus one.
The set of n for which P fails
has no least element.
Therefore the set is empty.

Pre-Chorus

(The indicator moves from red to green)
(File in triplicate: base, hypothesis, step)
(The bubble rests between the lines)
(Dust lifted on the finger: none)

Drop

Therefore, by induction.
Therefore, by induction.
Therefore, by induction.
For all n. For all n.
Quod erat demonstrandum.
Therefore, by induction.
Therefore, by induction.
For all n in the natural numbers.
For all n.
For all n.
Therefore, by induction.

Outro

No further cases remain.
The proof is closed.

Instrumental Outro