Cardinality of the set of all functions from N to N

[Flying_Goat]

Homework Statement

Let N^N be the set of all functions from N to N. Prove that |N^N|=c

Homework Equations

The Attempt at a Solution

I can prove that the set of all functions from N to {0,1} has cardinality of the continuum, but i can't generalise it. Any help would be appreciated.

 

[HallsofIvy]

Let f be a function from N to N. Construct a number, x, by writing a decimal point, then 0.f(1)f(2)f(3)... so that each function is mapped to the number, between 0 and 1, having the values of f as digits. That maps each function to a number. What numbers?

 

[Flying_Goat]

Thanks, I had thought of your argument when I was trying to prove |P(N)|=c but it didn't work out...I can't believe that it works for this question lol. Anyway thanks for your help.

 

[Flying_Goat]

Is the map injective? Because I could have f(2n-1)=12,f(2n)=3 or f(2n-1)=1,f(2n)=23 and both of these functions would give me 0.123123...