If $a$ and $b$ are linearly dependent, we write $\boldsymbol{b}=\lambda \boldsymbol{a}$ So we get: $$\langle \boldsymbol{a},\lambda \boldsymbol{a}\rangle^2=\lambda^2\langle \boldsymbol{a},\boldsymbol{a}\rangle^2=\lambda^2 ||\boldsymbol{a}||^4=||\boldsymbol{a}||^2||\boldsymbol{b}||^2.$$ So we have equality if $\boldsymbol{a}$ and $\boldsymbol{b}$ are linearly dependent. In the other case we look at the quadratic function $$||x\cdot \boldsymbol{a}+\boldsymbol{b}||^2=x^2||\boldsymbol{a}||^2+2x\langle\boldsymbol{a} ,\boldsymbol{b}\rangle + ||\boldsymbol{b}||^2.$$ This function is positive for every real $x$ if $\boldsymbol{a}$ and $\boldsymbol{b}$ are linearly independent. Thus it has no real zeroes, which means that $$\langle\boldsymbol{a}, \boldsymbol{b}\rangle^2 - ||\boldsymbol{a}||^2 ||\boldsymbol{b}||^2$$ is always negative. So we have: $$\langle\boldsymbol{a}, \boldsymbol{b}\rangle^2 < ||\boldsymbol{a}||^2||\boldsymbol{b}||^2,$$ which is the Cauchy-Schwarz inequality if $\boldsymbol{a}$ and $\boldsymbol{b}$ are linearly independent.