If we want to prove monotonically increasing, we only need to prove A2 > A 1 first, and then suppose AK+ 1 > AK, and prove AK+2 > AK+ 1, where k is an integer greater than or equal to 1.

In order to prove monotone reduction, we only need to prove A 1 > A2 first, then suppose AK > AK+ 1, and prove AK+ 1 > AK+2, where k is an integer greater than or equal to 1.

Related examples:

Example: {an} = {2 n} monotonically increasing.

Proof: the problem needs to be proved: a [n+1] > a[n]

(1) when n= 1, a [2] = 2 2 = 4 >; 2 = 2 1 = a [1], that is, the conclusion is established.

(2) Assuming that n=k, the conclusion holds, that is, A [k+1] > A[k], then when n=k+ 1

a[k+2]=2^(k+2)=2.2^(k+ 1)=2.a[k+ 1]>； 2.a[k]=2.2^k=2^[k+ 1]=a[k+ 1]

Therefore, the conclusion holds for all n, a [n+1] > A[n] is true, so {an} = {2 n} monotonically increases.