Convergent sequence of real numbers
By John Campbell •
Suppose that a_n is a convergent sequence of real numbers such that lim a_n = a > 0. Then there exists a K belongs to N such that a_n > 4a/5 holds for all n is greater than or equal to K. Is it true or false? How can i show it?
$\endgroup$ 04 Answers
$\begingroup$True.Take $\epsilon = a/5$.Can you continue?
$\endgroup$ $\begingroup$It is true.
$a_n$ is a convergent series.
For any $\epsilon >0$ there is an $N>0$ such that when $n>N, |a_n-a|<\epsilon$ let $\epsilon = \frac 15 a$
$\endgroup$ $\begingroup$Yes. Recall the definition of convergence of a sequence: what happens if $\epsilon = a / 5$?
$\endgroup$ $\begingroup$Hint: $\limsup_na_n=\lim_na_n=a$ by definition means that $a_n>a-\varepsilon$ if $n$ is large enough.
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