I'm looking for an easy read introductory book on analysis; ideally, something that comes together with exercises to solve (browny points if solutions are available) and that is written in English (or Swedish, but English is preferred). I'm not too worried about the rigor of proving every detail and am rather looking for something that builds intuition and explains the big ideas.
The background for this question is that my girlfriend is mentoring a school kid as part of a school program where kids go and experience life in the workplace for one week (she's a biology PhD). She keeps saying how smart the kid is and sent me a screenshot of his notes yesterday. Turns out that he, when left to his own devices, attempts to proof convergence limits for various series, which I think is not bad a hobby for a 14-year-old. I spoke with the teacher, and this kid will actually take a test next week to be allowed to attend math classes in the gymnasium (similar to the last years of prep school in the US), but considering all I've heard so far that might still get easy for him quickly.
Long story short, we are thinking of gifting him something to further his interest in math, while trying to avoid throwing him down the deep end. Any suggestions are highly appreciated :)
As I've already suggested in the comments, it can be a good idea to introduce strong students to mathematical competitions, so they can also get to know other young people with the same interests.
Along these lines, I will recommend the book The Art and Craft of Problem Solving, 3rd Edition by Paul Zeitz which not only is one of my favourites introducing general problem solving for math olympiads, but also has a fantastic calculus section in the last chapter. This book really teaches how to think, explains intuition and remains more than rigorous and above all his choice of topics are simply immensely fun.