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A bounded non-increasing sequence is a sequence of numbers, say {a_n}, such that the following two conditions hold: 1. The sequence is non-increasing, i.e., a_n+1 ≤ a_n for all n. 2. The sequence is bounded from below, i.e., there exists a real number b such that a_n ≥ b for all n.

A Real number s is the greatest lower bound of a set A if and only if: 1. s is a lower bound for A, i.e., ∀a ∈ A, a ≥ s. 2. s is greater than all other lower bounds of A, i.e., ∀l, if l is a lower bound of A, then l ≤ s.

We can prove that a bounded non-increasing sequence {a_n} converges to its GLB by following these steps: Let's consider the set A = {a_1, a_2, a_3, ...} containing all terms of the sequence {a_n}. Also, let b be the GLB of set A. - Since a_n+1 ≤ a_n for all n, every term a_n belongs to A. - Since set A is bounded from below, b, being its GLB, will exist. Now, we have to show that the limit of the sequence {a_n} is equal to b: lim(x→∞) a_n = b. Let ε > 0 be an arbitrary positive number. Since b is the greatest lower bound of A, we know that b + ε can't be a lower bound for A because b is the greatest one. Therefore, there exists an element a_N ∈ A such that a_N < b + ε. For sequence {a_n}, we know that a_n ≥ b for every n, so b is a lower bound for all a_n. Since a_N < b + ε and b ≤ a_n for all n, we have a_n ≤ a_N < b + ε for all n ≥ N. Now, let's consider |a_n - b|. Since a_n ≥ b, |a_n - b| = a_n - b. Hence, for all n ≥ N, we have: |a_n - b| = a_n - b < ε. This implies that the sequence {a_n} converges to b as its limit. Therefore, a bounded non-increasing sequence converges to its greatest lower bound.