For each positive integer $k$ define the function $f_k:[0,1] \rightarrow \mathbb{R}$ by $f_k(x)=\cos (\frac{x}{k})$ for $x$ in $[0,1]$. Show that the sequence of functions $\{f_k\}$ converges in the metric space $C([0,1],\mathbb{R})$.
To do this, would I just need to show that the sequence converges uniformly?It converges pointwise to $1$. Let $\epsilon>0$ Since $|\cos (\frac{x}{k})-1|\leq 1$ for all $k \in \mathbb{N}$ now I have to choose a $K$ in $\mathbb{N}$ such that $k \geq K \implies |\cos(\frac{x}{k})-1|< \epsilon$ for all $x \in [0,1]$. Now I am struggling to figure out what to do next, since I cannot just choose an $N$ that depends on $\epsilon$. Can I just choose $K>1$? Any help on how to proceed here?
$|\cos y-1| =2 \sin^{2} (\frac y 2) \leq \frac {y^{2}} 2$.