TITLE: Prove $\lim_{x\rightarrow 0}\cos (x)=1$ with the epsilon-delta definition of limits
QUESTION [7 upvotes]: Prove 
$$    
\lim_{x\rightarrow 0}\cos (x)=1
$$    
with the epsilon-delta definition of limits

REPLY [3 votes]: Note that $0\leq 1-\cos x=2\sin^2\frac{x}{2}\leq2|\sin\frac{x}{2}|\leq|x|$, then:
$\forall\epsilon>0,\exists\delta=\epsilon>0$, s.t. $\forall x, |x|<\delta, |1-\cos x|\left(\leq|x|<\delta\right)<\epsilon$.