If $1 > a-b > 0$ then $[a]geq[b]$If $a$ and $b$ are positive real numbers, then $a + b geq 2 sqrtab$.Proof if $n_k < n_k+1$ for all $k in mathbbN$, then $n_k geq k$ for all $k in mathbbN$.If $f(n) geq g(n)$ for all $n geq n_0$ then $f(n) geq cg(n)$ for all $n geq n_1$Inequality $2a^nb^nc^n+1geq a^2n+b^2n+c^2n$Suppose $0<a<b$. Prove for all $ngeq 2$, $0< sqrt[n]a< sqrt[n]b$.On some iterated inequalities and $x geq 5$Show that $2^n geq (n +2)^2$ for all $n geq 6$Induction for $3^ngeq n^2$What about $f$ if $fracdleft(xf(x)-xright)dx geq 0$Prove or disprove $|a| geq 2|b| implies |a+b| geq |b|$

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If $1 > a-b > 0$ then $[a]geq[b]$


If $a$ and $b$ are positive real numbers, then $a + b geq 2 sqrtab$.Proof if $n_k < n_k+1$ for all $k in mathbbN$, then $n_k geq k$ for all $k in mathbbN$.If $f(n) geq g(n)$ for all $n geq n_0$ then $f(n) geq cg(n)$ for all $n geq n_1$Inequality $2a^nb^nc^n+1geq a^2n+b^2n+c^2n$Suppose $0<a<b$. Prove for all $ngeq 2$, $0< sqrt[n]a< sqrt[n]b$.On some iterated inequalities and $x geq 5$Show that $2^n geq (n +2)^2$ for all $n geq 6$Induction for $3^ngeq n^2$What about $f$ if $fracdleft(xf(x)-xright)dx geq 0$Prove or disprove $|a| geq 2|b| implies |a+b| geq |b|$













-1












$begingroup$


what I can do for The case $[a]<[b]$.



I know that $[a]>[b]Rightarrow [a]geq [b]+1$



EDIT:



I have realized that the Initial assumption $[a]=[b]$ is false therefore I have changed it to $[a]geq [b]$










share|cite|improve this question











$endgroup$











  • $begingroup$
    For clarity, what does $[x]$ mean in this context? I've usually seen this to refer to the rounding function, i.e. rounding $x$ to the nearest integer, but a few counterexamples immediately come to mind in that instance
    $endgroup$
    – Eevee Trainer
    Mar 11 at 21:02










  • $begingroup$
    $[x]$ is the biggest integer for which we have $leq x$. Can you give me an counterexample?
    $endgroup$
    – New2Math
    Mar 11 at 21:04






  • 3




    $begingroup$
    What if $a=1.1$ and $b=0.9$?
    $endgroup$
    – clark
    Mar 11 at 21:06










  • $begingroup$
    Okay so you're referring to the floor function then @New2Math - so my counterexample would be irrelevant. (Were it rounding and not floor, I was thinking $a=0.6,b=0.4$.) clark's counterexample works for the floor function though.
    $endgroup$
    – Eevee Trainer
    Mar 11 at 21:10










  • $begingroup$
    @clark I see now that this assumption is false but the assumption $a-b>1Rightarrow [a]>[b]$ is true right, also If I would have changed the assumption in the question to $geq$ then it would be right, right?
    $endgroup$
    – New2Math
    Mar 11 at 21:10
















-1












$begingroup$


what I can do for The case $[a]<[b]$.



I know that $[a]>[b]Rightarrow [a]geq [b]+1$



EDIT:



I have realized that the Initial assumption $[a]=[b]$ is false therefore I have changed it to $[a]geq [b]$










share|cite|improve this question











$endgroup$











  • $begingroup$
    For clarity, what does $[x]$ mean in this context? I've usually seen this to refer to the rounding function, i.e. rounding $x$ to the nearest integer, but a few counterexamples immediately come to mind in that instance
    $endgroup$
    – Eevee Trainer
    Mar 11 at 21:02










  • $begingroup$
    $[x]$ is the biggest integer for which we have $leq x$. Can you give me an counterexample?
    $endgroup$
    – New2Math
    Mar 11 at 21:04






  • 3




    $begingroup$
    What if $a=1.1$ and $b=0.9$?
    $endgroup$
    – clark
    Mar 11 at 21:06










  • $begingroup$
    Okay so you're referring to the floor function then @New2Math - so my counterexample would be irrelevant. (Were it rounding and not floor, I was thinking $a=0.6,b=0.4$.) clark's counterexample works for the floor function though.
    $endgroup$
    – Eevee Trainer
    Mar 11 at 21:10










  • $begingroup$
    @clark I see now that this assumption is false but the assumption $a-b>1Rightarrow [a]>[b]$ is true right, also If I would have changed the assumption in the question to $geq$ then it would be right, right?
    $endgroup$
    – New2Math
    Mar 11 at 21:10














-1












-1








-1


1



$begingroup$


what I can do for The case $[a]<[b]$.



I know that $[a]>[b]Rightarrow [a]geq [b]+1$



EDIT:



I have realized that the Initial assumption $[a]=[b]$ is false therefore I have changed it to $[a]geq [b]$










share|cite|improve this question











$endgroup$




what I can do for The case $[a]<[b]$.



I know that $[a]>[b]Rightarrow [a]geq [b]+1$



EDIT:



I have realized that the Initial assumption $[a]=[b]$ is false therefore I have changed it to $[a]geq [b]$







inequality






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 11 at 21:16







New2Math

















asked Mar 11 at 20:59









New2MathNew2Math

11812




11812











  • $begingroup$
    For clarity, what does $[x]$ mean in this context? I've usually seen this to refer to the rounding function, i.e. rounding $x$ to the nearest integer, but a few counterexamples immediately come to mind in that instance
    $endgroup$
    – Eevee Trainer
    Mar 11 at 21:02










  • $begingroup$
    $[x]$ is the biggest integer for which we have $leq x$. Can you give me an counterexample?
    $endgroup$
    – New2Math
    Mar 11 at 21:04






  • 3




    $begingroup$
    What if $a=1.1$ and $b=0.9$?
    $endgroup$
    – clark
    Mar 11 at 21:06










  • $begingroup$
    Okay so you're referring to the floor function then @New2Math - so my counterexample would be irrelevant. (Were it rounding and not floor, I was thinking $a=0.6,b=0.4$.) clark's counterexample works for the floor function though.
    $endgroup$
    – Eevee Trainer
    Mar 11 at 21:10










  • $begingroup$
    @clark I see now that this assumption is false but the assumption $a-b>1Rightarrow [a]>[b]$ is true right, also If I would have changed the assumption in the question to $geq$ then it would be right, right?
    $endgroup$
    – New2Math
    Mar 11 at 21:10

















  • $begingroup$
    For clarity, what does $[x]$ mean in this context? I've usually seen this to refer to the rounding function, i.e. rounding $x$ to the nearest integer, but a few counterexamples immediately come to mind in that instance
    $endgroup$
    – Eevee Trainer
    Mar 11 at 21:02










  • $begingroup$
    $[x]$ is the biggest integer for which we have $leq x$. Can you give me an counterexample?
    $endgroup$
    – New2Math
    Mar 11 at 21:04






  • 3




    $begingroup$
    What if $a=1.1$ and $b=0.9$?
    $endgroup$
    – clark
    Mar 11 at 21:06










  • $begingroup$
    Okay so you're referring to the floor function then @New2Math - so my counterexample would be irrelevant. (Were it rounding and not floor, I was thinking $a=0.6,b=0.4$.) clark's counterexample works for the floor function though.
    $endgroup$
    – Eevee Trainer
    Mar 11 at 21:10










  • $begingroup$
    @clark I see now that this assumption is false but the assumption $a-b>1Rightarrow [a]>[b]$ is true right, also If I would have changed the assumption in the question to $geq$ then it would be right, right?
    $endgroup$
    – New2Math
    Mar 11 at 21:10
















$begingroup$
For clarity, what does $[x]$ mean in this context? I've usually seen this to refer to the rounding function, i.e. rounding $x$ to the nearest integer, but a few counterexamples immediately come to mind in that instance
$endgroup$
– Eevee Trainer
Mar 11 at 21:02




$begingroup$
For clarity, what does $[x]$ mean in this context? I've usually seen this to refer to the rounding function, i.e. rounding $x$ to the nearest integer, but a few counterexamples immediately come to mind in that instance
$endgroup$
– Eevee Trainer
Mar 11 at 21:02












$begingroup$
$[x]$ is the biggest integer for which we have $leq x$. Can you give me an counterexample?
$endgroup$
– New2Math
Mar 11 at 21:04




$begingroup$
$[x]$ is the biggest integer for which we have $leq x$. Can you give me an counterexample?
$endgroup$
– New2Math
Mar 11 at 21:04




3




3




$begingroup$
What if $a=1.1$ and $b=0.9$?
$endgroup$
– clark
Mar 11 at 21:06




$begingroup$
What if $a=1.1$ and $b=0.9$?
$endgroup$
– clark
Mar 11 at 21:06












$begingroup$
Okay so you're referring to the floor function then @New2Math - so my counterexample would be irrelevant. (Were it rounding and not floor, I was thinking $a=0.6,b=0.4$.) clark's counterexample works for the floor function though.
$endgroup$
– Eevee Trainer
Mar 11 at 21:10




$begingroup$
Okay so you're referring to the floor function then @New2Math - so my counterexample would be irrelevant. (Were it rounding and not floor, I was thinking $a=0.6,b=0.4$.) clark's counterexample works for the floor function though.
$endgroup$
– Eevee Trainer
Mar 11 at 21:10












$begingroup$
@clark I see now that this assumption is false but the assumption $a-b>1Rightarrow [a]>[b]$ is true right, also If I would have changed the assumption in the question to $geq$ then it would be right, right?
$endgroup$
– New2Math
Mar 11 at 21:10





$begingroup$
@clark I see now that this assumption is false but the assumption $a-b>1Rightarrow [a]>[b]$ is true right, also If I would have changed the assumption in the question to $geq$ then it would be right, right?
$endgroup$
– New2Math
Mar 11 at 21:10











1 Answer
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0












$begingroup$

If $a geq b$ then it immediately follows that $lfloor a rfloor geq lfloor brfloor$. So we don't even need the upper bound.



(All this says is that the floor function is monotone increasing.)






share|cite|improve this answer









$endgroup$












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    1 Answer
    1






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    active

    oldest

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    0












    $begingroup$

    If $a geq b$ then it immediately follows that $lfloor a rfloor geq lfloor brfloor$. So we don't even need the upper bound.



    (All this says is that the floor function is monotone increasing.)






    share|cite|improve this answer









    $endgroup$

















      0












      $begingroup$

      If $a geq b$ then it immediately follows that $lfloor a rfloor geq lfloor brfloor$. So we don't even need the upper bound.



      (All this says is that the floor function is monotone increasing.)






      share|cite|improve this answer









      $endgroup$















        0












        0








        0





        $begingroup$

        If $a geq b$ then it immediately follows that $lfloor a rfloor geq lfloor brfloor$. So we don't even need the upper bound.



        (All this says is that the floor function is monotone increasing.)






        share|cite|improve this answer









        $endgroup$



        If $a geq b$ then it immediately follows that $lfloor a rfloor geq lfloor brfloor$. So we don't even need the upper bound.



        (All this says is that the floor function is monotone increasing.)







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Mar 11 at 21:27









        Daniel McLauryDaniel McLaury

        15.9k32981




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