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Can anyone help me prove these problems?

If {$|a_n|$} is a divergent sequence, can {$a_n$} be a convergent sequence?

If {$a_n^2$} is a convergent sequence, can {$a_n$} be a divergent sequence?

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    $\begingroup$ What does ** mean $\endgroup$ – Jakobian Oct 18 '18 at 14:42
  • $\begingroup$ sorry, I haven't been very especifil, it means raised to the power of 2 $\endgroup$ – Lily island Oct 18 '18 at 14:49
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If {|$a_n$|} is a divergent sequence, can {$a_n$} be a convergent sequence?

This is false. As by a theorem: if the sequence {$a_n$} converges to A, then {|$a_n$|} converges to |A|.

The contrapositive states: If {|$a_n$|} is divergent then {$a_n$} is divergent.

If {$a_n^2$} is a convergent sequence, could {$a_n$} be a divergent sequence?

Consider $a_n=(-1)^n$

the square converges while the actual sequence does not

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For the first part, try to prove by contradiction. That is, assume that $\{a_n\}$ converges to some limit $a$ and show that $\{|a_n|\}$ converges. What will the limit of the latter sequence be?

For the second part, try to construct a sequence that repeatedly "hops" between two values, say $-1$ and $1$.

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