Example of cauchy sequence

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one of example of cauchy sequence show that

= 1/n - 1/(n+k)

and In the above we have used the inequality

1/(n+m)^2 <= ( 1/(n+m-1) - 1/(n+m) ) => i don't under stand where this come from

and what is inequality? can you give other example?
 
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  • #2
jbunniii
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$$\frac{1}{n+m-1} - \frac{1}{n+m} = \frac{1}{(n+m-1)(n+m)}$$
Assuming ##n## and ##m## are positive, it's clear that the right hand side is larger than ##1/(n+m)^2##, because ##n+m-1 < n+m##.
 
  • #3
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one of example of cauchy sequence show that

= 1/n - 1/(n+k)

and In the above we have used the inequality

1/(n+m)^2 <= ( 1/(n+m-1) - 1/(n+m) ) => i don't under stand where this come from

and what is inequality? can you give other example?
I'm not very clear about what the question is. Do you need to know how to prove the sequence is Cauchy? Do you need to understand what an inequality is?
 

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