Consider the following sequence of real numbers: Xsubn = (-1)^n + 1/n Assume that a does not equal +or-1. Prove that a is not a subsequential limit of Xsubn. Solution Let f: R--->R and let c be element in R. Show that the lim from x to c of f(x)=L if and only if lim from x to 0 of f(x+c)=L (if and only if: go both ways) b. Use either the epsilon- delta definition (which states: Let A be a subset of the reals and let c be a cluster point of A. For a function f: A--->R, a real number L is said to be a limit of f at c if, given any epsilon>0 there exists a delta>0 such that if x is an element of A and 0Reals(R) and let c be a cluster point of A, then the following are equivelent i. lim as x goes to c of f=L and ii. for every sequence Xsubn in A that converges to c such that Xsubn does not equal c for all n element N.

1. Consider the following sequence of real numbers:
Xsubn = (-1)^n + 1/n
Assume that a does not equal +or-1.
Prove that a is not a subsequential limit of Xsubn.
Solution
Let f: R--->R and let c be element in R. Show that the lim from x to c of f(x)=L if
and only if lim from x to 0 of f(x+c)=L (if and only if: go both ways) b. Use either the epsilon-
delta definition (which states: Let A be a subset of the reals and let c be a cluster point of A. For
a function f: A--->R, a real number L is said to be a limit of f at c if, given any epsilon>0 there
exists a delta>0 such that if x is an element of A and 0Reals(R) and let c be a cluster point of A,
then the following are equivelent i. lim as x goes to c of f=L and ii. for every sequence Xsubn in
A that converges to c such that Xsubn does not equal c for all n element N