Convergence of the maximal root of a family of quadratic polynomials as a parameter goes to 0

Question

I'm facing the following problem and I don't know how to solve it.

I have a quadratic polynomial $$ P_\lambda(z) = a_\lambda z^2 + b_\lambda z + c_\lambda, $$ depending on a parameter $\lambda \in [\lambda_0, +\infty)$, where $\lambda_0 >0$.

To fix the ideas, assume we know the following:

1. $a_\lambda >0$ and $c_\lambda <0$. In particular, $P_\lambda$ has a unique positive root for every $\lambda$.
2. Calling $z_\lambda$ the root of $P_\lambda$, I know that $$\lim_{\lambda \to +\infty} z_\lambda = 0,$$ implying that $$z_\star := \sup_{\lambda \in [\lambda_0,+\infty)} z_\lambda < +\infty.$$

(Of course, this means that the supremum is actually assumed for a $\lambda_\star \in [\lambda_0, +\infty)$) Now comes the fun part. We consider a perturbation of $P_\lambda$, $P_\lambda^\omega$, such that $$ P_\lambda^\omega = (a_\lambda + a(\omega))z^2 + (b_\lambda + b(\omega))z + c_\lambda + c(\omega) $$ where $a(\omega), b(\omega), c(\omega)$ are three $C^\infty$ functions of $\omega$ such that $$ \lim_{\omega \to 0} a(\omega) = \lim_{\omega \to 0} b(\omega) = \lim_{\omega \to 0} c(\omega) = 0. $$ For every fixed $\omega$ small enough, I can repeat the analysis above and show that $$z_\star^\omega := \sup_{\lambda \in [\lambda_0,+\infty)} z_\lambda^\omega < +\infty.$$ I would like to prove that $$ \lim_{\omega \to 0} z_\star^\omega = z_\star $$

What I've found up to now:

1. For a fixed $\lambda$, we certainly have $z_\lambda^\omega \to z_\lambda$. This is due to the continuity of the roots of a polynomial with respect to the coefficients.
2. If we see $z_\lambda$ as a function of $\lambda$, the result would be implied if we could show that $z_\lambda^\omega \to z_\lambda$ uniformly. In particular, by the above point, we know we have pointwise convergence.
3. The big problem seems to be that $[\lambda_0, +\infty)$ is unbounded. Indeed, I'm pretty sure a famous result by Brohnstein covers the case in which $\lambda \in \mathcal C$, where $\mathcal C$ is a compact subset of $\mathbb R$ (besides, on a compact set we have uniform continuity, which helps a lot for this problem).

Do you have any suggestion on how I could approach this problem? (Of course I can provide the actual coefficients if it is of any help)

Thank you!