Math Help - countable sets

1. countable sets

Hello! I need some help with these problems...
1. a. Prove that the set of all strictly increasing sequences of natural numbers, is not a countable set.
b. What can you say about the set of converging sequences of natural numbers? Is it a countable set, or not?
c. The same question about the set of periodic sequences of natural numbers.
2. Find out a function f:R->R, f injective, f not surjective.

Thank you!

2. Re: countable sets

For question 2, have you thought about the exponential function $e^{x}$?

3. Re: countable sets

Originally Posted by ely_en
1. a. Prove that the set of all strictly increasing sequences of natural numbers, is not a countable set.
b. What can you say about the set of converging sequences of natural numbers? Is it a countable set, or not?
For a)
Is it true that the power set of natural numbers is uncountable?
Is the set of all infinite subsets of natural numbers uncountable?
Can each infinite subset of natural numbers be considered a strictly increasing sequence of natural numbers?

For b)
Can there be a convergent sequence of natural numbers?

4. Re: countable sets

1.a) you can use cantor's diagonal argument for this.

suppose we have a surjection from N to the set of strictly increasing sequences of natural numbers. thus:

$1 \to n_{11},n_{12},n_{13},\dots$
$2 \to n_{21},n_{22},n_{23},\dots$
$3 \to n_{31},n_{32},n_{33},\dots$, etc.

now consider the following sequence: $n_{11}+1,n_{11}+n_{22}+1,n_{11}+n_{22}+n_{33}+1, \dots$

this is clearly strictly increasing, but it does not occur anywhere in our surjection.

1.b) it seems to me any convergent sequence of natural numbers is constant after a finite number of terms.

1.c) think about rational numbers, here.