Science:Math Exam Resources/Courses/MATH220/April 2005/Question 05
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Question 05 

Let be real numbers. Prove, using induction, that for all , You may assume the Triangle Inequality in the form for all real numbers x and y. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Use the fact that . 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. In the base case when , we have , so the statement is true. For the inductive step, we assume that holds for some (fixed) . Then we show that it also holds with replaced by : The first inequality is true by the triangle inequality and the second by the induction hypothesis. This finishes the inductive step and so the result is true for all by induction. 