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Novice

Registered: 2023-11-26

Posts: 1

Proving a function is bounded.

Hi so my question is asking to prove if 1/x is bounded when x belongs to the interval [1,infinty).This is the answer I came up with: Let M = 1 then (1 ≤ x) = (|1| ≤ x) = (-x ≤ 1 ≤ x) = (-x/x ≤ 1/x ≤ x/x) = (-1 ≤ 1/x ≤ 1) = (|1/x| ≤ 1) for all values of x on the interval for  [1,infinty).

This is my proof, if the proof is wrong then tell me what I did wrong, also the wording or the order of the proof might be weird so please help me with that. I am trying to learn real analysis. Thank you.

amnkb

Member

Registered: 2023-09-19

Posts: 253

Re: Proving a function is bounded.

Hi so my question is asking to prove if 1/x is bounded when x belongs to the interval [1,infinty).This is the answer I came up with: Let M = 1 then (1 ≤ x) = (|1| ≤ x) = (-x ≤ 1 ≤ x) = (-x/x ≤ 1/x ≤ x/x) = (-1 ≤ 1/x ≤ 1) = (|1/x| ≤ 1) for all values of x on the interval for  [1,infinty).

This is my proof, if the proof is wrong then tell me what I did wrong, also the wording or the order of the proof might be weird so please help me with that. I am trying to learn real analysis.

you start w/ M=1 so steps are
M≤x
|M|≤x
-x≤M≤x
-x/x≤M/x≤x/x
-1≤M/x≤1
|M/x|≤1
i dont know if this proves it
maybe dont restrict value of M
f(1)=1 is finite
maybe show f(x)=1/x is getting smaller on [1,infinity)
(by using deriv. maybe?)
show f(x)>0 on [1,infinity)
then f(x) bounded between 0 & 1
(these r just my thoughts)
(if somebody posts something different then theyre probably right- go with their ans.)