Why, once her grandmother is defeated, does the narrator feel "triumphant yet strangely saddened"?

The narrator felt 'triumphant, yet strangely saddened' after winning the argument because she had hurt her grandmother and made her sad.

According to the story, the narrator's grandmother had taken her into a part of the gully she had never visited before; the purpose of the trip was to allow the narrator to see a very tall palm tree situated in a clearing amidst the densest foliage.

When the narrator's grandmother asked whether there was anything taller than the palm tree in New York City, the narrator asserted that the buildings in the city were 'hundreds of times' taller than the palm tree. In fact, she maintained that The Empire State Building was taller than the palm tree because it had over a hundred floors. When the grandmother became exasperated, the narrator argued that the Empire State Building was even taller than Bissex Hill, a hill she had previously visited on the island.

When the grandmother became angry at the narrator, the narrator clinched the argument by crowing that she could send her grandmother a postcard to prove her claims. At this announcement, the grandmother knew she was beaten. The text tells us that the 'stubborn light' in her eyes had then dimmed, and she had 'turned and started back through the gully, walking slowly, her steps groping and uncertain, as if she were suddenly no longer sure of the way...' So, the narrator's victory was bitter-sweet; she had won the argument, but she had also succeeded in humiliating and hurting her grandmother.