Prove a function is in $L^2[0,1]$

A possible solution steps:

1. Prove that $$ \int_0^1\frac{1}{|x-t|^{1/2}}\,dt=2\sqrt{x}+2\sqrt{1-x}\le 2\sqrt{2}. $$
2. Prove that $\|g\|_\infty\le 2\sqrt{2}\|f\|_\infty$ (simple estimation by 1).
3. Prove that $\|g\|_1\le 2\sqrt{2}\|f\|_1$ (using e.g. Tonelli's theorem and 1).
4. Conclude that $\|g\|_2\le 2\sqrt{2}\|f\|_2$ by the Riesz-Thorin theorem.