Is every sequence convergent to zero belongs to one of $l_P$ for $p>1$?

Question

Is every sequence convergent to zero belongs to one of $l_P$ for p>1?

i.e Is $c_0\subset\cup_{p>0} l_p$ ?

We know that every $l_p$ sequence is in $c_0$. Now i am looking for converse.

I am not able to prove it?

Also i am not able to find examples.

Answer 1

No, this is not correct. Consider for example $a_n=\frac{1}{\ln(n+1)}$.