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It seems pretty obvious that it follows. But I can not give a good proof to it.

Let be $k\ge2$ an integer and {$a_n$} a sequence. Prove that if {$a_{kn+i}$} converges to $a$ for every $i\in$ {0,1,2,...,k-1}. Then {$a_n$} converges to $a$.

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Fix $\epsilon>0$. By you hypothesis, since all the $ \{a_{kn+i}\}_{n \in \mathbb{N}}$ converge to $a$, there exist $j_i, i=1, \ldots, k-1$ such that $|a_{kn+i}-a|<\epsilon$ whenever $n>j_i$. Now let $j$ be the largest of the $j_i$: you have that $|a_m-a|<\epsilon$ whenever $m>k i_j+k$. By the generality in the choice of $\epsilon$ the thesis follows.

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