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.

edit

My appeal to authority in the following answer makes no sense, because it seems Auslander's flows are actually assumed to be invertible. My own intuition is still that the issue is the same, so I'll keep this for now.

original

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.

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 know 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.

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.