Real Analysis Answers

Show that f =o(1) as x->a if and only if f(x)->0 as x->a.

The statement "If f = o(g) then f = O(g)" can be abbreviated o⇒O. Prove this.

Show that K(x-a) = O(k-a) [this is the letter O no the number 0] for any K.

If K is a constant, show that K(x-a) = o(x-a) [yes that is a "o" not a "0"] if and only if K=0.

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

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