Real Analysis Answers

23 Let A and B be subsets of a universal set U. Prove the following.
(a). A\B = (U\B) \(U\A)

Let X be a nonempty set , and let f and g defined on X and have bounded ranges in R. show that
Sup{f(x) + g(x):x element X}<= Sup {f(x):x element X} + Sup {g(x):x element X}
and that,
inf{f(x):x element X} + inf {g(x): x element X} <= inf { f(x) + g(x): x element X}.
also give example to show that each of these inequalities can be either equalities or strict inequalities.

Let S be a nonempty bounded set in R.
a.) let a>0, and let aS := (as:s element S). prove that, inf (aS)= a inf S, and sup (aS)= a sup S
b.) let b<0, and let bS := (bs:s element S). prove that, inf (bS)= b sup S, and sup (bS)= b inf S.

Show that a set of real numbers E is bounded if and only if there is a positive number r so that absolute value x<r for all x in e.

Show that absolute value of X={x,-x}

Let (x sub n) := & 1/ln(n+1) for n element of natural numbers.&
1. use the definition of limit to show that lim(x sub n)=0&
2. find the specific value of k(e) as required in the definition of limit for each of & (i) e = 1/2 and & & (ii) e=1/10.

For any[i] b [/i]element in real numbers, prove that [i]lim (b/n)=0[/i]

The first five terms of a sequence (x sub n) are given below.&
& 1/2,-1/4,1/8,-1/16,.....

assuming that the "natural pattern" indicated by these terms persist, give a formula for the nth term (x sub n).&

The sequence (x[sub]n[/sub]) is defined by the following formulas for the n[sup]th[/sup] term. Write the first five terms in each case.&
& (x[sub]n[/sub])= ((-1)[sup]n[/sup])/n

If [i]a,b[/i]& are element of real numbers, show that [b]|a+b|=|a|+|b|[/b] & if [b]ab>=0[/b]