For me, the intuition behind this is that you find a single pair $f(a)$ and $f(b)$ such that $a \ne b, f(a) \ne f(b)$. You have \begin{align} \left|f(a) - f(b)\right| \le (a-b)^2. \end{align}

Given the midpoint $m$ between $a$ and $b$, you also have \begin{align} \max\lbrace\left|f(a) - f(m)\right|, \left|f(m) - f(b)\right|\rbrace \le (a-m)^2. \end{align}

The left side is at most twice as small as the left side of the original equation, and the right side is exactly four times as small. Repeatedly subdividing, eventually the equation must be false.

For me, the intuition behind this is that you find a single pair $f(a)$ and $f(b)$ such that $a \ne b, f(a) \ne f(b)$. You have \begin{align} \left|f(a) - f(b)\right| \le (a-b)^2. \end{align}

Given the midpoint $m$ between $a$ and $b$, you also have \begin{align} \max\lbrace\left|f(a) - f(m)\right|, \left|f(m) - f(b)\right|\rbrace \le (a-m)^2. \end{align}

The left side is at least twice as small as the left side of the original equation, and the right side is exactly four times as small. Repeatedly subdividing, eventually the equation must be false.