For any real number x- -x-n - n- - where - - denotes GIF-

We have learnt the property that for any real number x, [x] gt;n ∀ n∈ℤ⇒ x≥ n+1. ...

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Question

For any real number $x, [x] > n \forall n \in Z \Rightarrow$ , where $[.]$ denotes GIF.

x = n + 1

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x \leq n + 1

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x \geq n + 1

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Solution

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x, [x] \leq n \forall n \in Z \Rightarrow

[.]

[x] \geq n \Leftrightarrow x \geq n,

n \in Z, x \in R

[.]

x < n \forall n \in Z,

[.]

[x] \leq n \forall n \in Z .

n

x = 6.6.

[.]

n \geq 1

a_{n} = [\frac{n}{[\sqrt{n}]}]

[x]

x

x

n

(1, 2, 3, . . . . . .2010)

a_{n} > a_{n + 1}

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