Let $f: \Bbb R \to \Bbb R$ be a differentiable function such that $\sup_{x \in \Bbb R}|f'(x)| \lt \infty$. Then

Solution 1:

All are correct.

1. If $\{x_n\}_{n\in\mathbb N}\subset\mathbb R$ is bounded, i.e., $\lvert x_n\rvert\le M<\infty$, then $$ \lvert\,f(x_n)-f(x_1)\rvert=\lvert x_n-x_1\rvert\lvert f'(y_n)\rvert, $$ for some $y_n\in(x_1,x_n)$, by virtue of the Mean Value Theorem, and hence $$ \lvert\,f(x_n)\rvert\le \lvert\,f(x_1)\rvert +\lvert x_n-x_1\rvert\lvert f'(y_n)\rvert\le \lvert\,f(x_1)\rvert +2M \|f'\|_\infty. $$

2. If If $\{x_n\}_{n\in\mathbb N}\subset\mathbb R$ is Cauchy, then $$ \lvert\,f(x_m)-f(x_n)\rvert=\lvert\,f'(y_{m,n})\rvert\lvert x_m-x_n\rvert\le\|f'\|_\infty \lvert x_m-x_n\rvert, $$ hence $\{f(x_n)\}_{n\in\mathbb N}$ Cauchy.

3. If $x_n\to x$, then $$ \lvert\,f(x_n)-f(x)\rvert=\lvert\,f'(y_n)\rvert\lvert x_n-x\rvert\le\|f'\|_\infty \lvert x_n-x\rvert, $$ where $y\in(x,x_n)$, and hence $f(x_n)\to f(x)$.

4. If $x,y\in\mathbb R$, then $$ \lvert\,f(x)-f(y)\rvert\le\|f'\|_\infty \lvert x-y\rvert, $$ etc...