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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