Does a real sequence (Xn)(X_n) with all terms between 00 and 1/n1/n have to eventually be decreasing?

Take a real sequence (Xn)(X_n) such that for any nn in N\mathbb{N}, 00≤xnx_n≤1/n1/n. At some point, does every term in the sequence have to be less than or equal to the prior one? I think that it does but am unsure of how to prove it.

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No. In fact, a sequence could increase to 1n\frac{1}{n}, like ak=1n−1n+ka_k = \frac{1}{n} – \frac{1}{n+k}. This is in fact an increasing sequence.
– ذ°رپر‚ذ¾ذ½ ذ²ر–ذ»ذ»ذ° ذ¾ذ»ذ¾ر„ ذ¼رچذ»ذ»ذ±رچر€ذ³
2 days ago

  

 

@ ذ°رپر‚ذ¾ذ½ ذ²ر–ذ»ذ»ذ° ذ¾ذ»ذ¾ر„ ذ¼رچذ»ذ»ذ±رچر€ذ³ Ah, right. Thanks.
– CuriousKid7
2 days ago

  

 

you are welcome.
– ذ°رپر‚ذ¾ذ½ ذ²ر–ذ»ذ»ذ° ذ¾ذ»ذ¾ر„ ذ¼رچذ»ذ»ذ±رچر€ذ³
2 days ago

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1 Answer
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No, take for example:
xn=1n−|sin(n)|2n=2−|sin(n)|n.x_n = \frac{1}{n} – \frac{|\sin(n)|}{2n} = \frac{2 – |\sin(n)|}{n}.