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A direct exponential minimization construction of a bounded pricing kernel proves the reverse implication without using a course theorem. ConsiderBoundedness of $P$ makes $F$ finite, continuous, and differentiable. On every bounded set of $h$, the exponential and its derivatives have deterministic bounds, so differentiation under expectation is justified.

We first prove coercivity: $F(h)\to \infty$ as $|h|\to \infty$. Otherwise there would be a sequence $h_j$ with $|h_j|\to \infty$ along which $F(h_j)$ stays bounded above. Pass to a subsequence with $h_j/|h_j|\to u$, $|u|=1$. There are two cases.

If $\mathbb{P}(u\cdot P<0)>0$, choose $a>0$ with $\delta =\mathbb{P}(u\cdot P\le -a)>0$. The bound on $P$ gives $h_j\cdot P\le -a|h_j|/2$ on this event for all large $j$. ThereforeIf instead $u\cdot P\ge 0$ almost surely, the portfolio condition forces $u\cdot p>0$, since $u\ne 0$. Hence $h_j\cdot p\to \infty$, and the nonnegative exponential term again forces $F(h_j)\to \infty$. Both cases contradict the chosen sequence.

A minimizing sequence is consequently bounded. It has a convergent subsequence, and continuity of $F$ gives a global minimizer $h_{\ast }$. At a minimizer each directional derivative vanishes, soThusIf $|P|\le L$, then $e^{-|h_{\ast }|L}\le Y\le e^{|h_{\ast }|L}$: the constructed pricing kernel is bounded and even bounded away from zero. The compactness, coercivity, and differentiation arguments above supply the needed proof rather than appealing to the fundamental theorem of asset pricing.