Does Adding a Constant to Two Real Numbers Preserve Their Relative Inequality?

[ice109]

if

$$s^{'}_{n} = s_n + b$$

and

$$S^{'}_{n} = S_n + b$$

and

$$s_n < A < S_n$$

does that imply that

$$s^{'}_{n} < A+b < S^{'}_{n}$$

?

ahh why can't i delete, i think this is probably obvious

the actual point of contention is if A is the only number between $s_n$ and $S_n$
is A+b the only number between $s^{'}_{n}$ and $S^{'}_{n}$ ? Surely it is but does it obviously follow?

 

[rock.freak667]

If you add b to everything you will get the last inequality. So it doesn't really imply it but you can get it by addition of b.

 

[Hurkyl]

A is the only number between $s_n$ and $S_n$
is A+b the only number between $s^{'}_{n}$ and $S^{'}_{n}$ ? Surely it is but does it obviously follow?

Could you be more precise in what you're asking?

 

[ice109]

Could you be more precise in what you're asking?

$s_n$ and $S_n$ are the inner and outer rectangles of an integral of f(x) and $s^{'}_{n}$ and $S^{'}_{n}$ are similarly for f(x)+b. I want to use the results of the proof of the first integral for the second one.

 

[HallsofIvy]

Yes, as rockfreak667 said, if you start with $s_n< A< S_n$ and add b to each part, $s_n+ b< A+ b< S_n+ b$ so $s'_n< A+ b< S'_n$.

If you want to be more specific, separate $s_n< A< S_n$ into $s_n< A$ and $A< S_n$. Adding b to each side of those, $s_n+ b< A+ b$ and $A+ b< S_n+ b$ so $s'_n< A+ b$ and $A+ b< S'_n$ which combine to $s'_n< A+ b< S'_n$.

 

[Hurkyl]

$s_n$ and $S_n$ are the inner and outer rectangles

How are you ordering rectangles? Or did you mean s_n and S_n are their height? So I presume A, s_n, and S_n are all supposed to be real numbers? Then (no matter what their values are!) it cannot possibly be true that A is the only number between s_n and S_n.