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

MATH220 December 2010

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Question 10 (b)
Let $k,\ell \in \mathbb {N}$. Prove, using the definition of convergence, that ${\frac {1}{kn+\ell }}\to 0$

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 $\displaystyle \{s_{n}\}_{n=1}^{\infty }$ of positive numbers converges to 0 if for all $\displaystyle \epsilon >0$ there exists a natural number n such that for all natural numbers $\displaystyle m\geq n$ you have that $\displaystyle s_{m}<\epsilon$.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Let $\displaystyle s_{n}$ be the sequence defined by $\displaystyle s_{n}={\frac {1}{kn+\ell }}$. Recall that a sequence $\displaystyle \{s_{n}\}_{n=1}^{\infty }$ of positive numbers converges to 0 if for all $\displaystyle \epsilon >0$ there exists a natural number n such that for all natural numbers $\displaystyle m\geq n$ you have that $\displaystyle s_{m}<\epsilon$. Pick an $\displaystyle \epsilon >0$. Then choose n a natural number so large such that $\displaystyle {\frac {1}{\epsilon }}<n$. Then for all $\displaystyle m\geq n$, we have that $\displaystyle {\frac {1}{\epsilon }}<n\leq m\leq km+\ell$ Hence $\displaystyle s_{m}={\frac {1}{km+\ell }}<\epsilon$ and thus the sequence converges to 0.

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