Proof in textbook regarding the uncountability of the set of all functions from $\mathbb{Z}^{+}$ to the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Question

In her proof of this problem, Epp defines a function that I do not understand. How is it possible for a function that sends each positive integer $n$ to $a_n$ to have codomain $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. My understanding of this function is that it would map $2$ to $0.02$. Could someone please explain to me what exactly this function does?

The function $F$ is defined for numbers in $[0,1]$. – copper.hat May 03 '20 at 03:26 — copper.hat, May 03 '20 at 03:26
$(F(0.a_1 a_2...) )(n) = a_n$. – copper.hat May 03 '20 at 03:35 — copper.hat, May 03 '20 at 03:35

score 2 · Accepted Answer · answered May 03 '20 at 03:14

Consider $x = 0.1358$, then $F(0.1358)(1) = 1$, $F(0.1358)(2) = 3$, $F(0.1358)(3) = 5$, $F(0.1358)(4) = 8$, and $F(0.1358)(n)$ for $n \geq 5$ is equal to zero.

The key thing to understand is that $F$ evaluated at a number in $[0,1]$ is itself a function.

Proof in textbook regarding the uncountability of the set of all functions from $\mathbb{Z}^{+}$ to the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

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