uniform convergence of $f_n(x) = \min (\cos x, 1 - \frac{1}{n}), x \in \mathbb{R}$

Question

Does $f_n(x) = \min (\cos x, 1 - \frac{1}{n}), x \in \mathbb{R}$ converge uniformly?

I have established that the pointwise limit is $$f(x) = \lim_{n \rightarrow \infty} f_n(x) = \begin{cases} 1, & \text{if}\ \cos x=1 \\ \cos x, & \text{otherwise} \end{cases}$$

Now I don't really know how to find the sup norm, $||f_n - f||_{\infty}$ to check if it goes to zero and the convergence is uniform.

I tend to struggle when the function has two parts and/or the limit has two parts, so any hints on dealing with these is helpful!

Answer 1

Yes, the convergence is uniform, since\begin{align}\left|\cos(x)-\min\left\{\cos x,1-\frac1n\right\}\right|&=\cos(x)-\min\left\{\cos x,1-\frac1n\right\}\\&=\begin{cases}0&\text{ if }\cos(x)\leqslant\frac1n\\\cos(x)-\left(1-\frac1n\right)&\text{ otherwise.}\end{cases}\\&\leqslant1-\left(1-\frac1n\right)\\&=\frac1n.\end{align}

Please note that your function $f$ is simply the co-sine function.