$$ \Big|\lang\bm u,\bm v\rang\Big|^2=\Big|\sum_{i=1}^nu_i^\dagger v_i\Big|^2\le \left(\sum_{i=1}^n|u_i^\dagger v_i|\right)^2\\=\left(\sum_{i=1}^n|\sqrt{u_i^\dagger v_iu_i^\dagger v_i}|\right)^2=\left(\sum_{i=1}^n|\sqrt{u_i^\dagger u_i}\sqrt{v_i^\dagger v_i}|\right)^2=\\\left(\sum_{i=1}^n|u_i|| v_i|\right)^2\le \left(\left(\sum_{i=1}^n|u_i|^2\right)^{\frac 12}\left(\sum_{i=1}^n|v_i|^2\right)^{\frac 12}\right)^2 =\lang\bm u,\bm u\rang\cdot\lang\bm v,\bm v\rang $$