To prove that $|\sqrt{a} - \sqrt{b}| \leq \sqrt{|a - b|}$, we'll start by squaring both sides of the inequality and then show that the inequality holds.

Step 1: Start with the squared version of the inequality

Given the inequality: $|\sqrt{a} - \sqrt{b}| \leq \sqrt{|a - b|}$ we first square both sides: $(\sqrt{a} - \sqrt{b})^2 \leq |a - b|$

Step 2: Expand the left-hand side

Expand the left side: $(\sqrt{a} - \sqrt{b})^2 = a - 2\sqrt{a}\sqrt{b} + b$ So, the inequality becomes: $a - 2\sqrt{a}\sqrt{b} + b \leq |a - b|$

Step 3: Consider two cases based on the sign of $a - b$

Case 1: $a \geq b$

• In this case, $|a - b| = a - b$.
• The inequality becomes: $a - 2\sqrt{a}\sqrt{b} + b \leq a - b$ Subtract $a + b$ from both sides: $-2\sqrt{a}\sqrt{b} \leq -2b$ Divide by $-2$ (which flips the inequality): $\sqrt{a}\sqrt{b} \geq b$ This simplifies to: $\sqrt{a} \geq \sqrt{b}$ This is true because $a \geq b$ implies $\sqrt{a} \geq \sqrt{b}$.

Case 2: $a < b$

• In this case, $|a - b| = b - a$.
• The inequality becomes: $a - 2\sqrt{a}\sqrt{b} + b \leq b - a$ Subtract $b + a$ from both sides: $-2\sqrt{a}\sqrt{b} \leq -2a$ Divide by $-2$: $\sqrt{a}\sqrt{b} \geq a$ This simplifies to: $\sqrt{b} \geq \sqrt{a}$ This is true because $b \geq a$ implies $\sqrt{b} \geq \sqrt{a}$.

Conclusion

In both cases, the inequality holds. Therefore, we have proved that: $|\sqrt{a} - \sqrt{b}| \leq \sqrt{|a - b|}$

Would you like more details on any part of this proof? Here are some related questions you might find interesting:

1. How does the triangle inequality apply to square roots?
2. Can we extend this inequality to higher powers, such as $a^{\frac{1}{n}}$ and $b^{\frac{1}{n}}$?
3. How does this inequality relate to the concept of the metric in geometry?
4. What is the geometric interpretation of the inequality $\sqrt{a+b} \leq \sqrt{a} + \sqrt{b}$?
5. How does this proof change if we use complex numbers instead of real numbers?

Tip: When proving inequalities, consider squaring both sides to simplify expressions, but be mindful of maintaining the correct inequality direction.