Student exchange; the introduction part of my essay?

I have to write an essay to apply for a student exchange program. Do you think this introduction part is good enough? Could you please tell me what you think and how I can make it better? I may have some mistakes about meaning or grammar.

Probably you know that the universities that accept foreign students aim to create an international and cosmopolitan atmosphere. So have you ever thought why they purpose such a thing? Because they believe gathering different cultures together would supply the possibility of thinking in different ways and reaching success. I write with great interest in applying for the position of the student exchange program which would give me the chance to meet a fully new culture and make use of its advantages.

Thanks for paying attention!

You probably know (It is commonly known would be better) that universities that accept foreign students aim to create an international and cosmopolitan atmosphere. Have you ever thought about their purpose?I believe that is is because they believe that gathering different cultures together would supply the possibility of thinking in different ways and reaching success. I write with great interest in applying for the student exchange program which would give me the chance to meet new people from an entirely new culture.

Thank you very much. I will change it and write the whole essay as soon as possible. (:

Very impressive! Here are the changes I would make:
So , have you ever thought about why they adhere to such a purpose ? It is because they believe gathering different cultures together in order to supply the possibility of thinking in different ways and reaching success. I write with great interest in applying...

I like it.

Thanks a lot. I hope the teachers will like it too. ^^