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Problem 52: (A Convergent Sequence Has A Unique Limit)

Let $(a_n)$ be a convergent sequence of real numbers. Prove that $(a_n)$ converges to a unique real number.

Note: The general way to prove something is unique is to suppose that there are two of those things, and then prove they must be equal. We need to prove that if $(a_n)$ converges to both $A$ and $B$, then we must have $A=B$. The contrapositive of this statement may be easier to work with, or possibly a proof by contradiction instead.

The following pages link to this page.

Problem.AConvergentSequenceHasAUniqueLimit
Schedule.20180618
Schedule.20180620
Schedule.20180622
Schedule.AllProblems

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Page last modified on June 20, 2018, at 06:50 PM

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HomePage Schedule Pictures Writing Tips LaTeX Commands External Resources List of Problems List of Definitions List of Theorems List of Conjectures All Recent changes (use this to find a page that may have been lost)	Problem A Convergent Sequence Has A Unique Limit Please Login to access more options. Problem 52: (A Convergent Sequence Has A Unique Limit) Let $(a_n)$ be a convergent sequence of real numbers. Prove that $(a_n)$ converges to a unique real number. Note: The general way to prove something is unique is to suppose that there are two of those things, and then prove they must be equal. We need to prove that if $(a_n)$ converges to both $A$ and $B$, then we must have $A=B$. The contrapositive of this statement may be easier to work with, or possibly a proof by contradiction instead. The following pages link to this page. Problem.AConvergentSequenceHasAUniqueLimit Schedule.20180618 Schedule.20180620 Schedule.20180622 Schedule.AllProblems Here are the old pages. Edit Page History Source Attach File Backlinks List Group Page last modified on June 20, 2018, at 06:50 PM
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