Prove that $\frac{7}{12}<\ln 2<\frac{5}{6}$ using real analysis I studying in Real Analysis 2, but I have no idea how to solve this problem. My guess is to use Mean Value Theorem or a similar theorem? Could any one help me?
Thanks.

J.G.'s elegant answer deserves to be restated in the following simple (and perhaps more familiar) form: we can approximate the integral
$$\ln 2 = \int_2^4 \frac1x\, dx$$
by a Riemann sum splitting the interval [2,4] into two parts of width $1$.  Since $1/x$ is strictly decreasing on this interval, the left-Riemann sum is a (strict) overestimate, and the right-Riemann sum is a (strict) underestimate.  Thus:
$$\frac{7}{12} = \frac13 + \frac14 < \int_2^4 \frac1x\, dx < \frac12 + \frac13 = \frac56.$$
