kyuhunl wrote:If a>b>0, is b<2?
1) 1/a > 1/2
2) 1/a + 1/b = 1
$$a > b > 0\,\,\,\,\,\mathop \Rightarrow \limits^{:\,\,ab\,\, > \,\,0} \,\,\,\,\,\frac{1}{b} > \frac{1}{a}\,\,\,\,\left( * \right)$$
$$b\,\,\mathop < \limits^? \,\,2$$
$$\left( 1 \right)\,\,\,\frac{1}{a} > \frac{1}{2}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\frac{1}{b} > \frac{1}{2}\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,b < 2\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{SUFF}}.$$
$$\left( {**} \right)\,\,\,\,\,\frac{1}{b} > \frac{1}{2}\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,\,2b\,\, > \,\,0} \,\,\,\,\,2 > b$$
$$\left( 2 \right)\,\,\,1 = \frac{1}{a} + \frac{1}{b}\,\,\mathop < \limits^{\left( * \right)} \,\,\,\frac{2}{b}\,\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {***} \right)} \,\,\,\,\,\,b < 2\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{SUFF}}.$$
$$\left( {***} \right)\,\,1 < \frac{2}{b}\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,\,b\,\, > \,\,0} \,\,\,\,\,b < 2$$
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio. 
