Learning Japanese - 見てみてください。"見て" is Different from "みて"

Japanese follows English.　英語の後に日本語が続きます。

A non-Japanese coworker of mine asked me about the following sentence the other day. He's been studying Japanese for several months and doesn't understand the difference between the two "mite." The sentence contains "mite" twice, but their meanings are different:

見てみてください。Mite mite kudasai. (Please take a loot at it.)

The first mite (見て) is a verb and means to see, look at, etc. "見" is used to describe something to do with vision or view, such as 見学 (observation, tour), 見晴らし (view) and 政見 (political view). On the other hand, the second mite (みて) is a subsidiary verb and has nothing to do with vision, but means "to try." Also, this mite (みて) is usually written in hiragana characters.

The second "mite (みて)" can be combined with other verbs as follows:

やってみて。 Yatte mite.  (Give it a try.)
着てみて。 Kite mite.  (Try it on.)
食べてみて。Tabete mite. (Try it.)

I explained all these things to him and he seemed to have satisfied, but I was surprised at this question and want to share this experience with non-native Japanese speakers who may have the same question. This gave me a chance to look at my mother language from a new perspective!

先日外国人の同僚から不思議な質問をされました。「見てみてください」というときの、「見て」と「みて」は両方とも「見る」じゃないのか。そうだとしたらなぜ2回言うのか、というのです。

最初はただびっくりだったのですが、知恵を振り絞り、最初の「見て」は見る（look、see）の意味だけど、二つ目の「みて」は視覚とは関係なく、「試しにやる」という意味だということ、だから漢字でなく平仮名で書くのが普通だと説明しました。

聞かれるまで全く考えたことはありませんでしたがもっともなご質問。日本語について考えさせられました｡｡｡