Elementary property of uniform continuous functions

\mathbf{Problem:} Suppose f is an uniform continuous function on the whole \mathbb{R}^n, then there is a constant K such that

\displaystyle \sup\limits_{\mathbb{R}^n\times\mathbb{R}^n}\{f(x)-f(y)-K|x-y|\}<\infty

\mathbf{Proof:} By the assumption of f, we know \exists\,\delta>0 such that for any |x-y|\leq\delta, |f(x)-f(y)|<1 holds

Then if \displaystyle |x-y|=r, we can conclude that \displaystyle |f(x)-f(y)|<\frac r\delta+1.

Choosing \displaystyle K=\frac 1\delta,  we have \forall \,|x-y|=r,

\displaystyle f(x)-f(y)-K|x-y|<\frac r\delta+1-\frac 1\delta\, r<1

\text{Q.E.D}\hfill \square

\bf{Corollary:} Suppose f is an uniform continuous function on the whole \mathbb{R}^n, then there are constants K and C such that |f(x)|<C+K|x|.

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