I am working through the following exercise:
Let $f\colon \mathbb{R} \rightarrow \mathbb{R}$ be a Riemann integrable function on every $[a,b],$ and such that $\int_{-\infty}^\infty |f(x)|\mathrm{d}x < \infty.$ Show that the Fourier sine and cosine transforms exist. That is, for every $\omega \geqslant 0,$ the following integrals converge: $$ F^s(\omega) := \frac{1}{\pi} \int_{-\infty}^\infty f(t)\sin(\omega t)\mathrm{d}t, \text{ and } F^c(\omega):= \frac{1}{\pi}\int_{-\infty}^{\infty}f(t)\cos(\omega t)\mathrm{d}t.$$ Furthermore, show that $F^s$ and $F^c$ are bounded functions.
My idea for the proof is the following: Since $f$ is integrable on $[a,b],$ then $f(t)\sin(\omega t)$ is too and \begin{align*} \left| \frac{1}{\pi} \int_a^b f(t)\sin(\omega t)\mathrm{d}t \right| &\leqslant \frac{1}{\pi}\int_a^b \left|f(t)\sin(\omega t)\right| \mathrm{d}t \leqslant \frac{1}{\pi}\int_a^b |f(t)|\mathrm{d}t \end{align*}since $-1 \leqslant \sin(\omega t) \leqslant 1$ for all $t$. Taking $b \to +\infty$ and $a \rightarrow - \infty$, we get $\int_{-\infty}^\infty |f(t)|\mathrm{d}t$ which converges by assumption. Then use comparison to conclude that $F^s$ converges. Similarly, $F^c$ converges. Is this the correct way to proceed? Also, the problem quantifies that $\omega \geqslant 0,$ but I'm not exactly sure where that would be necessary in the proof -- maybe it has something to do with sine being odd?
Thank you.