My Try:
It is clear that xn is monotonically increasing. If we assume that the sequence converges to a then a=a+√|a|n2. Hence a=0. So, I was going to prove that the sup of the sequence is 0. But failed. Can somebody please help me to complete the proof
Answer
xn+1=(√xn+12n2)2−14n4<(√xn+12n2)2
so
√xn+1<√xn+12n2
hence
√xn+1−√x1<12n∑k=11n2<π212
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