Below is how I prove it.
Case 1: $a = 0$
- $0^2 < b^2$ where $b$ is a positive number.
- $0 < b \times b$
- A positive number times a positive number is always positive.
- It is true.
Case 2: $a > 0$
- $a < b \Rightarrow a + x = b$
- $a^2 < b^2 \Rightarrow a^2 < (a+x)^2$
- $a^2 < (a+x) \times (a+x)$
- $a^2 < a^2 + 2ax + x^2$
- $0 < 2ax + x^2$
- It is true because $a, x$ are positive numbers.
I was wondering a) if my prove is correct and b) if there are other straightforward way to prove this?