Is $f_n(x) = \cos \left( \sqrt{x^2 + \frac{1}{n} }\right)$ uniform convergence on $ [0,1]$?
Of course we have $f_n \rightarrow \cos(x)$ but how prove that $$\operatorname{sup} \left| \cos \left( \sqrt{x^2 + \frac{1}{n} }\right) - \cos(x) \right| \rightarrow 0 $$ or $$\operatorname{sup} \left| \cos \left( \sqrt{x^2 + \frac{1}{n} }\right) - \cos(x) \right|\not\rightarrow 0 $$ ?