Vrftjkry

I have difficulties with the consistency part of the following question.

The question: Let $Y_1, ldots, Y_n$ be a random sample from a Poisson distribution with parameter $lambda > 0$. One observes only $W_i = mathbb{1}_{Y_i>0}$ for $i = 1, ldots, n$. Compute the likelihood associated with the sample $(W_1, ldots, W_n)$ and the MLE in $lambda$. Show that it is consistent in probability. Is it consistent in MSE (mean squared error)?

My solution

1) The MLE part:

For all $i$, the distribution of $W_i$ is Bernoulli with parameter $ p = 1 - sumlimits_{k=1}^{infty}frac{e^{-lambda}lambda^k}{k!}$ (it follows that $p =1 - e^{-lambda}$). So the likelihood of the sample $(W_1, ldots, W_n)$ is:
begin{align}
L(W vert p) = prodlimits_{i=1}^n p^{w_i}(1-p)^{1-w_i} = p^{sumlimits_{i=1}^n w_i}(1-p)^{n - sumlimits_{i=1}^n w_i} = p^X(1-p)^{n-X}
end{align}
where $X = sumlimits_{i=1}^n w_i$, is the number of success in n trials. The log-likelihood is:
begin{align}
l(W vert p) = X log(p) + (n-X) log(1-p)
end{align}
which is maximised at $hat{p} = frac{X}{n} = frac{1}{n}sumlimits_{i=1}^n w_i = overline{W}$, which is the mean of $W$. The MLE for $lambda$ is then $hat{lambda} = - log(1- overline{W})$.

2) The consistency in probability part:

We need to show that $forall epsilon > 0$, the probability $mathbb{P}(verthat{lambda} - lambda vert > epsilon) rightarrow 0 $ as $n rightarrow infty$. We can write this as:
begin{align}
mathbb{P}(vert(hat{lambda} - lambda vert > epsilon) = mathbb{P}(hat{lambda} - lambda > epsilon) + mathbb{P}(-hat{lambda} + lambda > epsilon) = mathbb{P}(n(1-e^{-epsilon - lambda}) < X) + mathbb{P}(n(1-e^{epsilon - lambda}) > X)
end{align}
and now we can use that $X$ has a Binomial$(n,p)$ distribution. But it becomes unclear to me how to proceed. Possibly there is some easier approach that I am missing?

How to solve the last part of the question is again unclear to me. Any help would be appreciated!