Calculus/Analysis

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Question
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?

Answer
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.

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