Proof/disproof $f(x)=x+\cos \sqrt x $ is uniformly continuity on $[0, \infty ).$
My approach: By definition, for every $\epsilon > 0$ there exist a $\delta >0 $ such that $ |f(x_1)-f(x_2) | \leq \epsilon $ whenever $ |x_1-x_2| \leq \delta , \forall x_1 , x_2 \in [0, \infty )$ So, $$ | x_1+\cos \sqrt x_1 - x_2-\cos \sqrt x_2| \leq |x_1- x_2|+|-2\sin \frac{(\sqrt x_1-\sqrt x_2)}{2}\sin \frac{(\sqrt x_1+\sqrt x_2)}{2}|\leq |x_1- x_2|+ 2\frac{|(\sqrt x_1-\sqrt x_2)|}{2}\frac{|(\sqrt x_1+\sqrt x_2)|}{2} \leq \frac{3}{2}|x_1- x_2| \leq \frac{3}{2}\delta =\epsilon$$
as $\sin x \leq x, \forall x \geq 0.$
Hence the function is uniformly continuous on $[0, \infty )$.
Is my approach correct or I had a mistake somewhere?
Thanks in advance.