A student is asked to prove whether the following statement (*) is true or false: (*) For all real numbers $a$ and $b , | a + b | < | a | + | b |$ The student's proof is as follows: Statement (*) is false. A counterexample is $a = 3 , b = 4$, as $| 3 + 4 | = 7$ and $| 3 | + | 4 | = 7$, but $7 < 7$ is false. Which of the following best describes the student's proof? A The statement ( $*$ ) is true, and the student's proof is not correct. B The statement (*) is false, but the student's proof is not correct: the counterexample is not valid. C The statement (*) is false, but the student's proof is not correct: the student needs to give all the values of $a$ and $b$ where $| a + b | < | a | + | b |$ is false. D The statement (*) is false, but the student's proof is not correct: the student should have instead stated that for all real numbers $a$ and $b , | a + b | \leq | a | + | b |$. E The statement (*) is false, and the student's proof is fully correct.
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A student is asked to prove whether the following statement (*) is true or false:\\
(*) For all real numbers $a$ and $b , | a + b | < | a | + | b |$
The student's proof is as follows:
Statement (*) is false. A counterexample is $a = 3 , b = 4$, as $| 3 + 4 | = 7$ and $| 3 | + | 4 | = 7$, but $7 < 7$ is false.
Which of the following best describes the student's proof?
A The statement ( $*$ ) is true, and the student's proof is not correct.\\
B The statement (*) is false, but the student's proof is not correct: the counterexample is not valid.\\
C The statement (*) is false, but the student's proof is not correct: the student needs to give all the values of $a$ and $b$ where $| a + b | < | a | + | b |$ is false.\\
D The statement (*) is false, but the student's proof is not correct: the student should have instead stated that for all real numbers $a$ and $b , | a + b | \leq | a | + | b |$.\\
E The statement (*) is false, and the student's proof is fully correct.