I have the following problem:
Consider $F:= \mathbb{R}^{2} \rightarrow \mathbb{R}^{+}$ such that $F_{K}(K,L), F_{L}(K,L)>0$ and $F_{KK}(K,L), F_{LL}(K,L)<0$ for all $K,L$. It also holds that $F$ is homogeneous of degree one. Moreover, the Inada Conditions hold, i.e. \begin{equation}\lim\limits_{K \rightarrow 0}F_{K} = \lim\limits_{L \rightarrow 0}F_{L} = +\infty \end{equation} and \begin{equation} \lim\limits_{K \rightarrow \infty}F_{K} = \lim\limits_{L \rightarrow \infty}F_{L} = 0 \end{equation} Show that $F(0,L) = 0$ for all $L$.
Now, my attempt:
From the Euler Theorem, we know that $F$ is such that \begin{equation} F(K,L) = K\cdot F_{K}(K,L) + L \cdot F_{L}(K,L) \end{equation}
From the differentiability of $F$ and the fact that the partial derivatives are homogeneous of degree zero, we can write \begin{gather} F(0,L) = \lim_{K\rightarrow 0}F(K,L) = \lim_{K\rightarrow 0}\frac{F_{K}(K,L)}{1/K} + L \cdot \lim_{K\rightarrow 0} F_{L}\left(\frac{L}{K}\right)\\ \end{gather}
Using the Inada Conditions, $\lim_{K\rightarrow 0} F_{L}\left(\frac{L}{K}\right) = 0$ because it is as if $L$ was increasing, in this case because we are dealing with an homogenous function of degree zero. By L'Hopital's rule, we can compute \begin{equation} \lim_{K\rightarrow 0}\frac{F_{K}(K,L)}{1/K} = \lim_{K\rightarrow 0}\frac{F_{KK}(K,L)}{-1/K^2} = 0 \end{equation}
Hence, $F(0,L) = 0$ as disered.
My doubt is on the last step. I understand that the second derivative involved, although not constant over $K$, is bounded over $K$. So, I can take the given limit. Is it wrong? Any ideas of an alternative proof?