Limit of a sequence does not goes to zero

  • Thread starter Seydlitz
  • Start date
  • #1
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4

Homework Statement


Could you guys please verify this proof of mine? I want to show that a limit with particular property does not go to 0. It is part of the proof that when a sequence have an ever increasing term then the limit of the sequence is not 0.

The Attempt at a Solution



The ##\lim_{n \to \infty} a_n \neq 0## if ##|a_n|<|a_{n+1}|##

Suppose ##\lim_{n \to \infty} a_n = 0##, then there exist ##n>0##, such that ##|a_N| < \epsilon## when ##N>n##. Taking an arbitrary ##a_n## as ##\epsilon##, we can get ##|a_N| < |a_n|.## Because it also true that ##N=n+1>n##, we get ##|a_n|>|a_{n+1}|##. A contradiction.

Hence it is not possible for the limit to be 0.

Thank You
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,263
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Homework Statement


Could you guys please verify this proof of mine? I want to show that a limit with particular property does not go to 0. It is part of the proof that when a sequence have an ever increasing term then the limit of the sequence is not 0.

The Attempt at a Solution



The ##\lim_{n \to \infty} a_n \neq 0## if ##|a_n|<|a_{n+1}|##

Suppose ##\lim_{n \to \infty} a_n = 0##, then there exist ##n>0##, such that ##|a_N| < \epsilon## when ##N>n##. Taking an arbitrary ##a_n## as ##\epsilon##, we can get ##|a_N| < |a_n|.## Because it also true that ##N=n+1>n##, we get ##|a_n|>|a_{n+1}|##. A contradiction.

Hence it is not possible for the limit to be 0.

Thank You

That looks just fine to me.
 
  • #3
263
4
Ok thanks for your verification Dick! I'm quite happy to discover this by myself.
 

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