I also did not understand why $\delta$ is continuous in the answer of Nicolas Tholozan, maybe they will clarify and I'll delete this, but in the meantime, I would have guessed this has no known elementary proof (or at least didn't in '88...) for the following reason: The only proof I can see is through the following theorem on page 67 in Auslander's '88 book Minimal Flows and their Extensions (in the chapter on distal flows):

 > Let $(X, T)$ be a flow and let $x \in X$. Then there is an almost periodic point $x^*$ which is proximal to $x$.

Your result easily follows, because the almost periodic point will be in the eventual image. Auslander makes the following comment after the theorem:
 
"Every known proof of this theorem requires the use of a "large" product space (another proof will be given in the next chapter). It would be interesting to find a direct proof."

I don't see a reduction of this to the result you are after, so possibly yours is easier, but this was the only reasonable approach I could see. I also don't know if Auslander's comment has already been addressed somewhere, I am not an expert on distality by any means.