Science:Math Exam Resources/Courses/MATH220/April 2005/Question 08
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Question 08 

Suppose that is a sequence that satisfies Prove that exists. 
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 

Have you heard of Cauchy sequences? Try seeing if the sequence is Cauchy. If you haven't, first see if you should (the curriculum might have changed since 2005). 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We can show that the sequence is Cauchy. Let , then if n and m are larger than (that's the value of N, you can see from the argument below how we might have guessed it), then if m ≥ n, we have that And since Cauchy sequences of real numbers are always convergent, we know that the limit of this sequence exists. 