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To which of the following statements can the transfer principle be applied? If you think it can't be applied to a certain statement, try to prove that the statement is false for the hyperreals, e.g., by giving a counterexample.
- For any real numbers
and 
, 
.
- The sine of any real number is between -1 and 1.
- For any real number
, there exists another real number 
that is greater than 
.
- For any real numbers
, there exists
another real number 
such that 
.
- For any real numbers
, there exists a rational number 
such that 
. (A rational number is
one that can be expressed as an integer divided by another integer.)
- For any real numbers
, 
, and 
, 
.
- For any real numbers
and 
, either 
or 
or 
- For any real number
, 
.
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