Suppose $X$~$N(0,\sigma^{2})$. Then $\epsilon$ ~$N(0, \sigma^{2})$ where $\epsilon$ is the residuals of the model?

Question

Suppose that $X$ is a sample of $n$ observations where $n$ is very large. We want to predict $Y$ and fit a regression model using ($X$,$Y$).

Is the following statement a proposition? If not, why? This is not homework, rather a question out of curiosity...

Suppose $X$~$N(0,\sigma^{2})$. Then $\epsilon$ ~$N(0, \sigma^{2})$ where $\epsilon$ is the residuals of the model $Y = \beta_{0} + \beta_{1}X + \epsilon$.