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.

To the extent that arithmetic is part of real analysis, it suffices to note that 
$$3^7=2187\lt4096=2^{12}$$
and
$$2^6\cdot4^5=2^{16}=65536\lt100000=10^5=2.5^5\cdot4^5$$
The desired inequalities now follow from the fact that $2.5\lt e\lt3$.
