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Let D be the ordered set of all possible words( not just English words, all strings of letters of abitrary length) using the latin alphabet using only lower case letters. The order is the lexicographic order as in a dictionaly (e.g. aaa< dog < door). Let A be the subset of D containing the words whose first letter is 'a'( e.g. a ∈ A, abcd ∈ A). Show that A has a supremum and find what it is.

I have no idea to solve this question..

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    Is azzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz......... sup A ? – 2017-02-03

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Note that $b$ is an upper bound. Now show that if $\beta$ is another upper bound, then $b \le \beta$.

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    Why the downvote? – 2017-02-04
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    Thank you. But, I did not down vote. – 2017-02-06
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    @KwangiYu: You are welcome. I know. I have my own personal downvoter now :-). – 2017-02-07