This weekend, American cyclist Lance Armstrong will make his final assault on the Tour de France, aiming for an unprecedented seventh straight victory. Not since Michael Jordan has an individual athlete so dominated a sport, but that is not what makes ordinary Americans care about his progress. Armstrong – who plans to retire after the tour ends on Sunday in Paris – has become a living symbol of individual triumph over illness and adversity.

Diagnosed with advanced testicular cancer at age 25 and given little chance for recovery, and no hope of ever cycling again, Mr. Armstrong tapped into an inner strength few people are aware they possess to become the ultimate survival story. This resonates not only with the estimated 9.8 million Americans living with cancer, but with folks who face smaller challenges in their own daily life. He makes us feel that we might be capable of the same.

This son of Texas was raised by a young working mother who always told him to make every obstacle an opportunity. That determination has been Lance Armstrong’s great strength as an athlete and as an individual. After the worst surgeries and chemotherapy were over, there were moments when he was tempted to give up on his chosen career and opt for a quieter life. But doubt does not last long in the “life or death, on or off” Mr. Armstrong worldview. With the increased appreciation of the opportunities of this life and determination to make the most of his own talent, he worked ceaselessly not only to return to the professional cycling tour, but to become better than he ever was before. Three years after his near-fatal illness, Mr. Armstrong came back to win the Tour de France for the first time. He has won it every year since.

As Mr. Armstrong recounted in his autobiography, “I had learned what it means to ride in the Tour de France. It’s not about the bike. It’s a metaphor for life. It’s not only the longest race in the world, but also the most exulting and heartbreaking and potentially tragic. It imposes every conceivable element to the rider and more: cold, heat, mountains, plains, ruts, flat tires, high winds, unspeakably bad luck, unthinkable beauty, yawning senselessness, and above all, a great deep self-questioning. During our lives we’re faced with so many different elements as well. We experience so many setbacks and fight such hand-to-hand battle with failure, head down in the rain, just trying to stay upright and have a little hope. The tour is not just a bike race, not at all. It is a test. It tests you physically, it tests you mentally, and even tests you morally.”

Watching Mr. Armstrong push himself on the mountain roads of the Alps, with his crucifix and a Texas-shaped medal bobbing against his chest, is a powerful reminder of not just physical but mental discipline, that shift where a moment of pain is transformed into an opportunity to excel. In a world where not enough people are taught to enjoy challenges, Mr. Armstrong’s example helps us to push ourselves in sickness and in health, to endure discomfort and gain a defiant pride and pleasure from overcoming it.

With the final weekend of the Tour de France upon us, Armstrong fever is at full pitch. A giant billboard outside New York’s Penn Station showing Mr. Armstrong with scars on his then-bald head from cancer surgery somehow brings courage and comfort to daily commuters. Fifty million yellow rubber “LiveStrong” bracelets have been sold for a dollar each to raise money for cancer research. They are so successful that they are in danger of overexposure, but their message is so essentially heartening that it must transcend a trend. His life story has become a national reminder about the power of the individual will. One admirer, Jay Vanderzee, explained Mr. Armstrong’s visceral appeal: “Lance is all about winning. He has inspired me and millions of others to take the extra step, read the extra line, get the second opinion, and to say no when anyone tells you to settle for what life puts in front of you. There is no price tag anyone can put on the value he has given us by the example of his actions.”

Amid all the applause, one element of Mr. Armstrong’s example has been too little remarked upon: The man is unapologetically ambitious, and he refuses to play anyone’s after school-special idea of a saintly cancer survivor. As the new book “Lance Armstrong’s War” makes clear, this is a man driven by the intense competitiveness of a control freak: the thirst for information, absolute precision, and obsessive preparation in the pursuit of his goal. This single-minded will to win recalls an unfashionable but essential element of the American character previously evangelized by folks like Vince Lombardi, General Patton, and even Richard Nixon. This seems to me not a bad thing.

This weekend offers a chance to watch excellence in action. After Sunday, we will know whether Lance Armstrong has succeeded in his attempt to win an unprecedented seventh Tour de France. In a larger sense, he has already succeeded by not only challenging himself, but by challenging the rest of us and thereby transforming our culture. At a time when we need heroes, Lance Armstrong’s message is that we need only to look within.