things i did:
1) if An converges to a, i can show that Bn converges to a , also.
similarly, if Bn converges to b, i can show An converges to b , also.
therefore, An converges iff Bn converges. and, obveriously, a=b
2) suppose Xn converges to a and let C = [m,n] then...
two question on analysis...
1）Prove : Let (An) amd (Bn) be sequences in a metric space S such that d (An , Bn) → 0. Then (An) converges if and only if (Bn) converges, and if they converge, they have the same limit.
2）Prove: Let C be a closed set and let (Xn) be a sequence in C converging to a...
The set A={1/n : n in N} is not compact.
A) prove this by explicitly finding a family of open sets which covers A but has no finite subfamily whcih also covers A.
B) Find another family of open sets which covers A and does have a finite subfamily which cobers A.
a. 1/n + 1/m : m and n are both in N
b. x in irrational #s : x ≤ root 2 ∪ N
c. the straight line L through 2points a and b in R^n.
for part c. i got: intA= empty ; bdA=clA=accA=L Is this correct? how about part a and part b...i am so confused...
1) give an example of a sequence (An) of open sets such that the intersetion
∩ An is not open.
n in N
2) If A is a nonempty set of real numbers bounded below with no minumum, then infA is an accumulation point of A.
Could somebody gives some hint on...