Consider a linear regression model, wherein: $$ y_{i}=x_{i}\beta+\epsilon_{i} $$ where notation is standard and $x$ is a scalar. Let us further impose the following restriction: $$ \epsilon_{i}|x_{i}\sim N(0,\sigma^{2}) $$
Given mean independence, the OLS estimator $\hat{\beta}_{OLS}$ is both consistent and unbiased. Invoking the concept of almost sure convergence, we have that: $$ \Pr\left(\lim_{n\rightarrow\infty}\hat{\beta}-\beta=0\right)=1 $$
My question is as follows: $\hat{\beta}$ is distributed exactly as normal, given our normality assumption of the error terms. The normal distribution is continuous, and as a result, the probability of the random variable $\hat{\beta}$ taking on a value of $\beta$ is exactly 0. How can/do I interpret convergence concepts in terms of continuous distributions? Does the above imply that the distribution of $\hat{\beta}$ becomes degenerate over time?