Proof of a limit point on a plane.

Prove that every infinite and bounded point collection in the plane (R2) has a limit point.

Working with the limit of Supremum.

Let {En} be a collection of non-empty sets. Show that LimSupEn={x: x is in En for infinitely many n}

Signed Measure

If v is a signed measure, E is v-null if |v|(E)=0

Cauchy Sequences

Suppose that {x_n} is a sequence which satisfies |x_{n+1} - x_n| <= 1/log n Is this sequence Cauchy? What about the one satisfying |x_{n+1} - x_n| <= 1/(1 + epsilon)^n where epsilon > 0?

Functions: Limits

Using the definition of a limit (rather than the limit theorems) prove that lim {x -> a+} f(x) exists and find the limit in each of the following cases a) f(x) = x/|x|, a = 0. b) f(x) = x + |x|, a = -1. c) f(x) = (x - 1)/(x^2 - 1), a = 1. In which cases do lim {x -> a-} f(x) and lim {x -> a} f(x) also exist?