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Q)set S is bounded,

Let r be any real number and t any positive real number

Let T = {r - tx : x e S}.

Show T is bounded and inf T = r - t sup S

This is what I have worked upon so far:

To show that T is bounded, I need to find a number m such that for all y in T, y >= m.

Since S is bounded above, there is a number M such that for all s in S, s <= M.

Can I show that m = r-tM is a lower bound for T?

Questioner:Akuma

Country:London, City of, United Kingdom

Category:Calculus

Private:No

Subject:Math- Real Analysis

Question:

set S is bounded,

Let r be any real number and t any positive real number

Let T = {r - tx : x e S}.

Show T is bounded and inf T = r - t sup S

This is what I have worked upon so far:

To show that T is bounded, I need to find a number m such that for all y in T, y >= m.

Since S is bounded above, there is a number M such that for all s in S, s <= M.

Can I show that m = r-tM is a lower bound for T?

.................................................

Analysis is the study of inequalities -- lots of fun.

OK, then, if S is bounded, there is M such that for all x in S, M >= x.

(and a similar statement for a lower bound)

So: M >= x

Now do some algebra:

M >= x

-tM <= -tx <<< multiply by -t, switching to <=

+r +r

--------------

r - tM <= r - tx

OK, then; r-tM is a lower bound of your T.

Now, you have shown that for every UB of S, there is a LB of T. You should have no trouble showing that if K is sup S, r - tK is inf T.

Do a similar thing with the lower bounds of S, etc.

Calculus

Answers by Expert:

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.**Education/Credentials**

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