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A set with partial order is the minimum requirement on the set, where
infimum is defined.
Infimum of a partial order set A is the greatest lower bound of element of
A. The word greatest lower bound explains the meaning of infimum. As you
are interested sequence and infimum, i will restrict to the case of R. Formal
definition is the following:
Definition: Let A be a subset of R. Then x ∈ R is called infimum of A if
1. For all a ∈ A, we must have x ≤ a.
2. If y ∈ R such that y ≤ a then we must have y ≤ x
If such x does not exists we say that infimum of A is −∞.
Example: Lett A = { 1n : n ∈ N}. Then we claim that inf A= 0.
We see that for all n ∈ N, 0 < 1n . So 1st condition of definition is satisfied.
Also there is some y, 0 < y then by Archimedean property, there is some n ∈ N
such that 0 < 1n < y. Hence if there is some y such that y <
1
n for all n then
we must have y ≤ 0. So 2nd condition of definition satisfies. Hence we see
Inf(A) = 0.
So for finding infimum of a set, we have to check both the condition of defi-
nition. In the following discussion, we will assume that inf(A)...