Does Mathematics research in proven problems have any importance? This is my first post in this community, please excuse me if this kind of post is inappropriate.

I completed my second year of my Bachelor's Degree in Mathematics (where I study, all Bachelor's Degrees are a 3 year course, unlike in USA).
I have asked one of my Professors to organize a Mathematics Research Internship for which I wouldn't get paid, as I just wanted to get into the world of research and improve my level of math, and hopefully get some experience in research.

As I am working towards the end of it, I am starting to wonder if this Internship will have any credibility or a positive effect on my CV once I apply to a Master's Degree program (I don't think anyone will look at what I am doing now when I apply to a PhD program). What provokes such doubts in me is the fact that I am doing "pseudo-research", basically I am trying to prove some properties that have been proved long ago. Most of the stuff that I proved can be most likely found in graduate Mathematics text books. I think that whoever will be studying my CV will have doubts about my credibility.

So my question, how will my internship's "scientific report" look (basically a presentation of what I have done, all the proofs and all the application I have done)?
It is not going look bad at all. It sounds like the Professor has given you a project that is achievable in the time frame, and satisfies your goals of improving your math and getting some experience in research. It should have a positive effect both on your CV and on your understanding of the research process.

In my PhD experience, much of the beginning was is exploring the topic, reading the literature, and trying to figure out a new direction. This can take many months, and can even change as your understanding improves or new information comes to light.

I don't think people will expect to see original research on new problems until the latter stages of your PhD.