Show uniform convergence of $f_n (x)=\sqrt {\sin \left(\frac {x}{n}\right)+\cos\left(\frac {x}{n}\right) }$

Question

How can I prove uniform convergence of $$ f_n (x) =\sqrt {\sin \left(\frac {x}{n}\right)+\cos\left(\frac {x}{n}\right)} $$ for $x \in \left[0,\frac{\pi}{2}\right]$

I have been able to show its pointwise limit is $1$.

Answer 1

Hint: Define $$f(x)= \sqrt{\sin(x) + \cos(x)}$$ This function is continuous and thus uniformly continuous on $[0; \frac{\pi}{2}]$.

Added: We have that for every $x\in [0; \frac{\pi}{2}]$ that $$ \left\vert \frac{x}{n} - \frac{x}{m} \right\vert = \left\vert \frac{1}{n} - \frac{1}{m} \right\vert \cdot \vert x \vert \leq \frac{\pi}{2} \left\vert \frac{1}{n} - \frac{1}{m} \right\vert $$ Furthermore, we have $$ \vert f_n(x) - f_m(x) \vert = \vert f(\frac{x}{n}) - f(\frac{x}{m}) \vert. $$ Of course, we don't need the uniform continuity (as shown in the comments by p4sch), but I consider it usefull to think this way. It allows one to also tackle different problems (like for example the uniform convergence of $g_n(x)=\sqrt{x^2+\frac{1}{n}}$ to the absolute value on the whole real line).

Answer 2

Hint: use the inequalities $|\sin(t)| \leq |t|$ and $|\cos(t)-1|\leq \frac{t^2}{2}$.