1) The Fibonacci series shall never yield non whole numbers; `double`s are unnecessary and possibly computationally expensive. Try a `uintmax_t`. It will probably give you values of up to 2^64 - 1 (18,446,744,073,709,551,615). That should certainly be sufficient. Additionally, I would put in code to terminate the program as it reaches the upper limit of `uintmax_t`.
 
2) Your could have a classic I/O problem: output buffering. Try to put a `\n` in every print statement to encourage proper flushing. So rather than having the <newline> come at the beginning of each print statement, try putting it at the end. If you like it the way it is, you can call `fflush(stdout)` after every `printf()`. While we're on the topic, your first two `printf()` statements can be combined. More info at [this](https://stackoverflow.com/q/1716296/12786236) SO question.  

3) Obviously the `Sleep()` functions are unnecessary to your code. I presume that they are there for the aesthetic value of the output, but I would take them out. Doing so would make your code OS-independent as well as neater. But obviously, depending on the assignment, appearance may be above all.  

4) Finally, despite the mathematical purpose of the program, `math.h` is not needed.  
  
Addendum: You asked how to make it more professional. You could make a Fibonacci function encapsulating the algorithmic properties of your code, while leaving the aesthetic properties (sleep, welcome message) in `main()`. Example: 
```
uintmax_t num = get_next_fibbonaci(/*reference to struct storing state || values necessary for calculation*/);
```