Take a real sequence (Xn)(X_n) such that for any nn in N\mathbb{N}, 00â‰¤xnx_nâ‰¤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}.