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Math Help - Countable set problem

  1. #1
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    Countable set problem

    Can somebody please give me some help on this analysis problem?
    Show that the set of all nondecreasing functions from N on to {0,1,2} is countable?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by namelessguy View Post
    Can somebody please give me some help on this analysis problem?
    Show that the set of all nondecreasing functions from N on to {0,1,2} is countable?
    First consider an increasing function from \mathbb{N} to \{0, 1, 2\} which takes all three
    values 0, 1, 2. This is charaterised by two natural numbers n_1 and n_2,
    where the function changes from 0 to 1, and from 1 to 2. The set of ordered
    pairs of naturals is countable (you should already know this, so the set of
    increasing functions from \mathbb{N} to \{0, 1, 2\} which take all three values is
    countable.

    Now the set of all functions that we seek is the union of those that take all
    three values as considered above, those that take two of the possible values.
    You need to show that this latter set of functions is also countable. Then use
    the result that the union of two countable sets is also countable.

    RonL
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