Science:Math Exam Resources/Courses/MATH220/December 2010/Question 10 (b)
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Question 10 (b) 

Let . Prove, using the definition of convergence, that 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Remember here that k and l are fixed in this problem. We want to know what happens as n tends to infinity. 
Hint 2 

The definition of convergence is the following: A sequence of positive numbers converges to 0 if for all there exists a natural number n such that for all natural numbers you have that . 
Checking a solution serves two purposes: helping you if, after having used all the hints, 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. Let be the sequence defined by . Recall that a sequence of positive numbers converges to 0 if for all there exists a natural number n such that for all natural numbers you have that . Pick an . Then choose n a natural number so large such that . Then for all , we have that
Hence
and thus the sequence converges to 0. 