Every function can $f \colon \mathbb R \to \mathbb R$ can be represented as the sum of two one-to-one functions

Question

How can one use the Principle of Transfinite Induction, i.e, "Let $P(z)$ be a mathematical statement that depends on the ordinal $z$. Suppose whenever $P(\eta)$ is true $\forall \eta<z,P(z)$ is also true. Then $P(z)$ is true for every ordinal $z$." To prove that every function

$$f:\mathbb{R} \to \mathbb{R}$$

can be represented by as the sum of two one-to-one functions?

Answer 1

Hint: Fix an enumeration $(x_i \mid i < 2^{\aleph_0})$ of $\mathbb R$.

Claim. There exists a sequence $(g_\xi \mid \xi \le 2^{\aleph_0})$ such that for all $\xi \le 2^{\aleph_0}$

• $g_\xi \colon \{x_i \mid i < \xi \} \to \mathbb R$ is a function,
• $g_\zeta \subseteq g_\xi$ for all $\zeta \le \xi$ and
• $f \restriction \{ x_i \mid i < \xi \} - g_\xi \colon \{x_i \mid i < \xi \} \to \mathbb R$ is one-to-one.

Then $g := g_{2^{\aleph_0}}$ is as desired.

Prove the claim by induction on $\xi \le 2^{\aleph_0}$. At limit steps $g_\xi$ is the union of all the previous defined functions. At successor steps you need to shift the value of $f(x_{\xi-1})$ to a value that hasn't already been used.