Let $(x_{n})_{n\geq 1}$ be a sequence of real numbers so:
i) $ x_{n+2}\geq [x_{n+1}] + \{ x_{n} \}$, $n\geq 1$, and
ii) $\lim\limits_{n \to \infty }(x_{n+1}-x_{n})=0$.
Prove that the sequence is convergent.
Convergence of a sequence with integer part and fractional part
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real-analysis
sequences-and-series
limits
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0what is $\{x_n\}$? – 2017-02-13
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0the fractional part of x_n – 2017-02-13
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0Note that $[x_{n+2}]\ge[x_{n+1}]$. When $x_{k+1}-x_k$ becomes very small, $[x_{k+2}]>[x_{k+1}]$ will violate the first inequality for $n=k+1$. Thus, $[x_n]$ is constant from some place. Then $i)$ is just $x_{n+2}\ge x_n$ for large enough $n$ and $x_n$ is bounded. Take supremums by odd and even indices and prove they cannot be different by $ii)$. – 2017-02-13
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0I can't reach at a final result using this. Can you please be more detailed? – 2017-02-13