I'm learning about Fourier analysis and need help with the following problem (which is part of a subchapter on $L^p$ spaces):

Using the Cauchy-Schwarz inequality show that if $f \in L^2[-\pi, \pi]$, then $(1)$ $f \in L^1[-\pi, \pi]$ and $(2)$ $\|f\|_{L^1} \leq \sqrt{2\pi} \|f\|_{L^2}$.

**My work and thoughts:**

$(1)$ We note that if $f \in L^2[-\pi, \pi]$, then the inequality $\|f\|_{L^1} \leq \sqrt{2\pi} \|f\|_{L^2}$ implies $\|f\|_{L^1} < \infty$ and hence $f \in L^1[-\pi, \pi]$.

What is bothering me about my proof for $(1)$ is that I'm using $(2)$ which I have not yet prove. Is my work correct and/or is there another way of showing $(1)$ without making use of $(2)$?

$(2)$ This is the part where I should apply the Cauchy-Schwarz inequality using the well-known "multiplication by one trick": $\int |f|^2 = \int g h$ where $g = |f|^2$ and $h = 1$. Since it is the first time I'm using the C-S inequality in the context of $L^p$ norms I'm having some difficulties applying it. Any help on this problem would be greatly appreciated.