Science:Math Exam Resources/Courses/MATH220/December 2010/Question 09 (a)

MATH220 December 2010

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Question 09 (a)
Let $a_{n}\to 1$. Prove that there exists $N$ such that if $n>N$ then $\displaystyle {}|a_{n}+1|<3$.

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 $\displaystyle \epsilon =1$ in the definition of convergence.

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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. By definition of convergence, there exists a number N such that for all $\displaystyle n>N$ we have that $\displaystyle |a_{n}-1|<1$ (in the definition of convergence, we took $\displaystyle \epsilon =1$ above). Thus, $\displaystyle {\begin{aligned}|a_{n}+1|&=|a_{n}-1+2|\\&\leq |a_{n}-1|+2\\&<1+2\\&=3\end{aligned}}$

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