Children chased the rainbow's end in vain.
Meaning: The children tried to reach the end of the rainbow, but they couldn't because it's not a real thing.
Choice of Words: Children are young people. Chased means to run after something to catch it. Rainbow's end is the point where the rainbow seems to touch the ground. In vain means without success or result. The sentence describes children trying to catch the rainbow, but it's impossible, as rainbows are optical illusions.
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Related Expressions