$f:[0,1]\to \mathbb R$ , be continuous function then prove that $$\int_0^1f^2(x)dx\geq \biggl(\int_0^1|f(x)| \biggr) ^2$$
I tried this for $x^2$
For that above is true
But I checked following proof Which is complete opposite to above. Proving the Cauchy-Schwarz integral inequality in a different way
Please help me to find that where is I am making wrong ?
Any help will be appreciated