I was thinking of using the definition of a limit but am not sure with how to start.
Let a(n) be a convergent sequence with limit L, and let K e N. Show that a(n+k) is also convergent with limit L
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limits
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1Hint: if $n\ge N\Rightarrow |a_n-L|<\epsilon$, then.....or better, note that $a_{n+k}$ is a subsequence of $a_n$. – 2017-01-29
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0What does "K e N" mean? – 2017-01-29
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0K is an element of the natural numbers – 2017-01-29
1 Answers
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Recall the definition of limit:
$b_{n} \to L$ if for all $\epsilon > 0$, there exists $M \in \mathbb N$, such that if $m > M$, then $|b_m - L| < \epsilon$.
We want to show that $a_{n+k} \to L$, right? So we need to start with some $\epsilon > 0$.
At this point, we know that $\lim a_{n} = L$. From definition, using the $\epsilon$ we started with, there is some $M \in \mathbb N$ such that $|a_{n} - L | < \epsilon$ whenever $n > M$. Now, if $n > M$, then $n + k > M+k$, right?
So, if $(n+k) > M+k$, then $|a_{n+k} - L| < \epsilon$ is true. So, our new $M$ is just $M+k$. So, for any $\epsilon$, we have found an $M$ satisfying the limit conditions. Hence, it follows that $a_{n+k} \to L$.