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uncountable or countable

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Please help with the following problem.

A) Let P be the set of all functions f : N -----> N such that for some M in N and all n>M, f(n + 1) = f (n). In other words, f is in P provided that after some point, f is constant. Show that P is countable.
b) Let E be the set of all strictly increasing functions f : N ----> N. Show that E is uncountable.

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This posting offers help to conclude that E is uncountable or countable. A step by step solution is provided.

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a) To show that a set is countable, set up a bijection between it and a countable set.
First, remember, that the set Q of rational numbers is countable.
Second, we will use the fact that a subset of a countable set is (at most) countable.
Also, remember that if X and Y are countable, then the cartesian product X x Y (the set of pairs (x,y)) is also countable.
We will establish a bijection between P and a subset of the cartesian product of Q and N.
Let f be a function from P. Then, there is a number M, such that f(n) = k, a natural number, for all n>M.
Write all values ...

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