\[(a + b)\left( \frac{1}{a} + \frac{1}{b} \right) \geq 4;\ \ \ \ \ \ a > 0,\ \ \ \ \ \]
\[b > 0\]
\[\frac{(a + b)(b + a)}{\text{ab}} \geq 4\]
\[\frac{(a + b)^{2}}{\text{ab}} \geq 4\]
\[\frac{a^{2} + b^{2} + 2ab - 4ab}{\text{ab}} \geq 0\]
\[\frac{(a - b)^{2}}{\text{ab}} \geq 0 \Longrightarrow верно.\]
\[ab > 0,\ т.к.\ a > 0,\ \ \ \ b > 0;\]
\[(a - b)^{2} > 0 \Longrightarrow всегда.\ \]