From right to left all digits are multiplied by a decreasing sequence starting with 9.

We can prove that is a decreasing sequence.

So is a decreasing sequence of positive integers.

As we are assuming, , the above equation is in a geometric decreasing sequence.

There are infinite decreasing sequences of sets under .

This equivalent sequence has a smaller biggest number, so eventually we’ll get to 0 since we have a decreasing sequence of positive numbers.

This process can be repeated infinitely, producing an infinite decreasing sequence of positive solution sizes.

Hence there can be no infinite monotonically decreasing sequence.

There's no infinite strictly decreasing sequence of positive integers).

If then where is a strictly decreasing sequence such that In a similar way if or if .

Following this idea for a decreasing sequence of h values ending in 1 is guaranteed to leave a sorted list in the end.

These conditions imply Noetherianity, which means that every strictly decreasing sequence of monomials is finite.

A monotonically decreasing sequence is defined similarly.

Let be a decreasing sequence of normal subgroups of of finite index.

So it must converge to some limit h. Similarly, f(2n+1) is a decreasing sequence which is bounded below.

The most natural ordering that has this property is lexicographic ordering of the decreasing sequence of their elements.

If is a monotonically decreasing sequence of plurisubharmonic functions then is plurisubharmonic.

If x is a decreasing sequence with intersection 0, then the sequence m(x) has limit 0.

The limit of a decreasing sequence of subharmonic functions is subharmonic (or identically equal to ).

Because the ordinals are well-ordering principle, there are no infinite strictly decreasing sequences of ordinals.

The set Kleene star of words over ordered lexicographically (as in a dictionary) is not a well-quasi-order because it contains the infinite decreasing sequence .

Every strictly decreasing sequence of elements of the set must terminate after only finitely many steps (assuming the axiom of dependent choice).

We say a decreasing sequence of submodules is an I-filtration if ; moreover, it is stable if for sufficiently large n.

Every continuous plurisubharmonic function can be obtained as the limit of a monotonically decreasing sequence of smooth plurisubharmonic functions.

Indeed, a theorem of Greenbaum, Pták and Strakoš states that for every monotonically decreasing sequence a, .