= Attainment of the limit of a monotone sequence
The <supremum> of an increasing <bounded sequence> need not be a <maximum> of its <set> of terms. For $a_n=1-1/n$, every term is strictly less than one; the <supremum> and <limit of a sequence> are one, but there is no <maximum>. Every candidate term is beaten by the next. The decreasing <sequence> $1+1/n$ similarly has <infimum> one without a <minimum>.
If an increasing <sequence> with <limit of a sequence> $L$ attains $a_k=L$, it must remain there: $L=a_k\leq a_n\leq L$ for $n\geq k$. The decreasing case is identical with inequalities reversed. Thus attainment of the <limit of a sequence> is equivalent to an eventually constant <tail of a sequence> for a convergent <monotone sequence>. The increasing <sequence> $0,1/2,1,1,\ldots$ has both <supremum> and <maximum> one. <Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl>[§2.1, Exercise 2.1.13: a convergent monotone sequence attaining its limit]{ref}
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