I haven't changed my argument. If you read above I clarified that by ''external'' I mean external not to one's body but to one's mind.

This pen in front of me that I am perceiving now is external to my body but it is not external to my mind. I now look away. ''The pen'' (if it exists) is now external to both my body and my mind. I cannot know whether it exists until I perceive it again, at which point it ceases to be external to my mind.

The idealist does not deny that things exist independently of one's mind, i.e. independently of one's perception.

The idealist does claim that one cannot know that things exist independently of one's mind since this would entail perception of unperceived objects, a contradiction in terms.

If the idealist accepts the existence of a mind-independent world, he does so on the basis not of reason but of faith.

The idealist doesn't deny that one is perceiving an object. What he denies is that one can know that that object exists independently of his perception of it.

The realist maintains that the object can exist independently of one's perception of it.

The idealist simply asks: How could you possibly know that?

But the realist position commits one to perceiving an unperceived object (i.e. ''seeing'' an object outside of one's mind), a contradiction in terms, hence refuted.

But that's precisely the point - the idealist's claim (original post) that the existence of the external world cannot be proved is irrefutable.

Perhaps I was not clear in what I meant by ''external''. By external I mean external not to one's body but to one's mind, i.e. to one's perception.

Your position would commit you to the existence of a thing (i.e. an external object) which you cannot experience.

How can you prove the existence of that which you cannot experience? As far as I am aware, no philosopher has ever succeeded in this.

Thanks for this. I found the following section, from Hegel's Science of Logic, also helpful:

Things are called “in themselves” in so far as abstraction is made from all being-for-other, which means simply, in so far as they are thought devoid of all determination, as nothings. In this sense, it is of course impossible to know what the thing in itself is. For the question: what? demands that determinations be assigned; but since the things of which they are to be assigned are at the same time supposed to be things in themselves, which means, in effect, to be without any determination, the question is made thoughtlessly impossible to answer, or else only an absurd answer is given. (SL, 121)

From what I understand, Hegel is saying that:

(i) A thing-in-itself would have to lack all determination.
(ii) But if it lacked all determination it would be nothing.
(iii) But if the thing-in-itself is nothing then it is not possible to know anything about it at all.

In other words, in the very act of conceiving of a ''thing-in-itself'' Kant is conceiving of that which cannot be conceived, a contradiction.

This brings to mind the maxim of the early Wittgenstein: ''whereof one cannot speak thereof one must remain silent'' (Tractatus).