Axiomatic WikiWWikiTTheoremsBasic properties of the triangular numbers
Theorem·T51
Triangular numbers strictly increase, and every value up to and including the current top fits strictly below the next one.
For every
and
with
:
and, for every
and
with
:
In words
For any natural and a second above it: the first's triangular number is below the second's. And for any natural and an amount no larger than it: adding the amount to that natural's triangular number still stays below the next triangular number.
Never needed: F05 · F10 · F13 · A02 · A03 · A04 · A05 · A07 · A09 (computed from the citation graph, not asserted).
Proof
- 1
- 2Monotonicity. By induction on : for the claim is vacuous (no ). Assume the claim for (every has , the induction hypothesis "IH"); show it for . Applying Gap with , (using ): , and since (L17, adding ), . Now let , i.e. (L15). If , this is exactly , just shown. If , the IH gives , and combined with by transitivity: .
- 3T05 concludes: the monotonicity claim holds for every , hence for every .
∎
Remarks
The gap property is what makes the pairing map
injective: each "diagonal"
contributes a block of
to the range of
, entirely below
, so distinct diagonals never collide, and within one diagonal,
is read straight off by subtracting
.