Prove that $\lim\limits_{n\to \infty} \cos(a+b/n)=\cos(a)$ by the definition of limits, where $a$ and $b$ are positive numbers My approach:
For all $\epsilon \gt 0$, we need to find $K$ in natural numbers, s.t.
$$|\cos(a+b/n)-\cos(a)| <\epsilon, \text{ for all } n\ge K.$$
I want to convert $|\cos(a+b/n)-\cos(a)| <\epsilon$ to a form that $n$ is in the L.H.S. and $\epsilon$ is in the R.H.S. in order to find the $K$.
I don’t know what to do for this step.

First, notice that $|\cos(x)-\cos(y)|\le |x-y|$. Take $x=a+b/n$ and $y=a$. Then,
$$|\cos(a+b/n)-\cos(a)|\le|a+b/n-a|=|b/n|=b/n$$
Then, for $n\ge K$ we have $1/n<1/K$. Could take $K$ such that $1/K<\epsilon/b$ and we're done.
