Real Analysis Answers

Let x_sub(1) = a>0 and x_sub(n+1) = x_sub(n) + 1/x_sub(n). Show that x_sub(n) diverges

Show that it is possible to have a sequence (x_sub(n)) diverge while the sequence (|x_sub(n)|) converges

Show that it is possible for a sequence to have no convergent subsequences at all.

suppose S is a nonempty open set that isn't the whole real line. Show that there is a sequence of elements of S that converges to an element of C(S).

a) give an example to show it is possible to have lim(x_sub(n) + y_sub(n)) exist without having lim x_sub(n) or lim y_sub(n) exist

b) give an example to show it is possible to have lim(x_sub(n) * y_sub(n)) exist without having lim x_sub(n) or lim y_sub(n) exist

Prove if |x_sub(n) - L| <= b_sub(n) and lim b_sub(n)=0, then lim x_sub(n)=L

Using the binomial theorem, find the limit of Sn=(1+1/n)^n-1

Task
If (a sub_n) and (b sub_n) are cauchy sequences, show directly that (a sub_n * b sub_n) is also a cauchy sequence.
Detailed explanation: NO
Specific requirements:
Prove

Prove: A subset S of the real numbers is closed if and only if lim (as n goes to infinity) X_n is an element of S whenever (X_n) is a convergent sequence whose terms are all in S.

what are dummy variables?