= Solution
If the bounded <injective> <Fourier transform> $L^1\to C_0$ were surjective, the <open mapping theorem> for <Banach spaces> would make its inverse bounded, yielding $\|h\|_1\leq C\|\widehat h\|_\infty$. But the preceding $g_n$ have <uniform norm> $2$, whereas $\|h_n\|_1$ diverges.
For a quantitative lower bound, on $0<x\leq1/4$ we have $\sin(2\pi x)\geq4x$. On each interval $I_j=[(j+1/12)/n,(j+5/12)/n]$, $|\sin(2\pi n x)|\geq1/2$. For $1\leq j\leq\lfloor n/8\rfloor$ these intervals lie below $1/4$ for large $n$, so
$$
\|h_n\|_1\geq\frac2{\pi^2}\sum_j\int_{I_j}\frac{dx}{x}
\geq\frac{2}{3\pi^2}\sum_{j=1}^{\lfloor n/8\rfloor}\frac1{j+5/12}\longrightarrow\infty.
$$
This contradicts boundedness of the inverse. Therefore \b[the <Fourier transform> has proper dense image in $C_0(\mathbb R)$].
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