Uniform continuity of $f(x) = \frac{x+1}{x+2}$ on $(-2,\infty)$

Question

Is the function $f(x) = \frac{x+1}{x+2}$ uniformly continuous on $(-2,\infty)$? I know how to prove this for the case when the domain is closed or is, say, $[-1, \infty)$, but I am not sure how to estimate the fraction from above when working with this domain.

Answer 1

Consider taking two sequences: ${x_n}= -2 + \frac {1}{n}$ and ${y_n}= -2 + \frac {1}{2n}$, from there argue by contradiction.