A foundation piece of self-esteem is a child’s growing sense of skills and abilities. Are they being challenged? Are they learning new things?
An easy way to build this in is giving challenges in play. If they are building with blocks, challenge them to build it taller. If they are climbing, challenge them to do it in a new way. If they are playing with play-doh, challenge them to make some new creation. As they rise to meet the challenge in play, they are learning to take on challenges in life.
Another way to provide this is to enroll them in classes that provide new levels of challenges as they progress. This would include sports, musical instruments, cooking classes and foreign languages.
For self esteem, it can be helpful to focus most on their individual progress and their skills rather than the competition.
Once they are school age, a version of this would be to have them teach you one new thing they learned in school each week. This is a challenge to remember something and be able to explain it in detail to you. For challenges to be beneficial in this way overtime, they don’t have to be big. These can be small challenges given regularly.
Parenting by Dr. Rene 
I am really working hard on my daughters self esteem and willingness to accept challenges but I don’t think I am very successful. Every little detail and losing will make her quit….How to teach them not to quit?
“Every little detail and losing will make her quit” sounds like you might be dealing with perfectionistic tendency or a bit of anxiety more than low self-esteem overall. While all of these things can be related, teasing out will help to direct how to best address her. Does it feel like perfectionism – a need to get things right, an upset over making mistakes, holding her own to higher standards than others? Does it feel like anxiety – is she anxious easily over other things, is she mostly worried about others seeing mistakes or how she is perceived?
Also how old is she?
Great little article. Solid self-esteem is bases on real achievement. Some excellent ideas. I especially like the idea of challenge in play, and the one about teaching you something they have learned.