When very young children show persistence as a trait, it's probably at least part of their nature.
When my daughter was a baby (in the age 1 range), she found a duck puzzle in the toy box. You had to fit four pieces together to make the duck. She sat down with it and struggled to put it together. Some people here may have observed that the idea of turning a puzzle piece to make it fit isn't always obvious to a young child, and many (most?) kids have to learn how to do this. She was getting stuck on this point. I didn't help her --- we used to call her "By-myself baby."
Anyway, she was struggling to do the puzzle for a while. It got to a point where I was worrying that she was going to throw it across the floor and start crying. Instead, she kept at it quietly and eventually put it together. I breathed a sigh of relief, happy that it was done.
Then. . .she dumped it out onto the floor and started all over again. She did that puzzle again and again until it was trivial for her.
You can't teach a baby to be persistent in this kind of situation. This was 100% internal drive. She continues to display this trait to this day (she's nearly 7). It doesn't surface in every situation, but when it does, she really tries. We encourage her, but the drive primarily comes from her. So in her case, there is way more nature at work. (YMMV)
