# If (an) ∞ n=1 is a converging sequence, and for all n ∈ Z +, an ≤ 1, then limn→∞ an ≤ 1

by mrmathboi   Last Updated September 21, 2018 03:20 AM

The question I want to answer is : If ($$a_n$$) is a converging sequence, and for all n ∈ Z +, $$a_n$$ ≤ 1, then $$lim_{n→∞}a_n$$ ≤ 1. Here is my proof for this.

Proof: Suppose ($$a_n$$) is converges to some limit a, and for all n ∈ Z +, $$a_n$$ ≤ 1, and further suppose, absurdly, that $$a>1$$. Then $$a-1>0$$ and so we set $$\epsilon=a-1 >0$$. Then there exists $$N_1\in$$ N such that for all $$n \in$$ N, if $$n>N_1$$, then |$$a_n−a$$|< $$a-1$$. This implies that $$(a-1)-a$$ < $$a_n$$ implies that $$1 < a_n$$ which contradicts the assumption that $$a_n$$ ≤ 1. QED

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