# Using Cauchy-Schwarz to prove the triangle inequality

Using the following Cauchy-Schwarz Inequality for integrals,

$$\int_c^d\int_a^bf^2(x,y)dxdy\int_c^d\int_a^bg^2(x,y)dxdy \geq \Bigg[\int_c^d\int_a^bf(x,y)g(x,y)dxdy\Bigg]^2$$

Prove: $$\sqrt{\int_c^d\int_a^b[f(x,y) + g(x,y)]^2dxdy} \leq \sqrt{\int_c^d\int_a^bf^2(x,y)dxdy} + \sqrt{\int_c^d\int_a^bg^2(x,y)dxdy}$$

I am struggling with this proof. I tried starting by expanding the left handside and constructing an inequality by a substitution of the C-S inequality but this did not seem to help.

Putting double integrals seems superfluous so lets use just one. We use this form of Cauchy-Schartz (which is just Hölder for $$p=q=2$$): \begin{align*} \left(\int fg\right)^2 \leq \int f^2 \int g^2. \end{align*}
It is equivalent to prove the square of the equality: \begin{align*} \int (f+g)^2 \leq \int f^2 + \int g^2 + 2\sqrt{\int f^2} \sqrt{\int g^2}. \end{align*}
Note that we may expand the LHS and apply Cauchy-Schwartz on the cross-term: \begin{align*} \int (f+g)^2 = \int f^2 + \int g^2 + 2\int fg \leq \int f^2 + \int g^2 + 2\sqrt{\int f^2}\sqrt{\int g^2}. \end{align*}