Are there infinitely many numbers in the interval [0,1]?
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This is part of a series on common misconceptions.
True or False?
There are infinitely many numbers in the interval \( [0, 1]. \)
Why some people say it's true: Every new combination of digits after "0." leads to a new number between 0 and 1. Since there are infinitely many possible combinations, there are infinitely many numbers in \([0, 1]\).
Why some people say it's false: Since the length of the interval is finite, there can't possibly be infinitely many numbers in \( [0, 1] \).
The statement is \( \color{green}{\textbf{true}}\).
Proof: The fraction \( \frac{ 1 } { n } \) will always be in the interval \( [0, 1 ] \) for all positive integers \( n \). Since there are infinitely many positive integers, there are infinitely many numbers in the given interval. \( _\square\)
Rebuttal: (Address any concerns that people have)
Reply: (Explain why the argument is not vald)
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