Show that the set of functions from the positive integers to the set {0. 1. 2, 3, 4. 5. 6. 7. 8, 9} is uncountable. [Hint: First set up a one-to-one correspondence between the set of real numbers between 0 and 1 and a subset of these functions. Do this by associating to the real number 0. the function f with f(n) = dn.]

Step 1</p>

We have to show that the set of functions from the set of positive integers to the set

{ 0,1,2,3,4,5,6,7,8,9} is uncountable.

Step 2</p>

Proof

Let F be the set of all functions from the set of positive integers to the set {0,1,2,3,4,5,6,7,8,9}

We have to prove F is uncountable.