Recent content by jdcasey9

Homework Statement Let {fn} be a sequence of measurable functions defined on a measurable set E. Define E0 to be the set of points x in E at which {fn(x)} converges. Is the set E0 measurable? Homework Equations Proposition 2: Let the function f be defined on a measurable set E...

Homework Statement Prove that E is measurable if and only if E \bigcap K is measurable for every compact set K. Homework Equations E is measurable if for each \epsilon < 0 we can find a closed set F and an open set G with F \subset E \subset G such that m*(G\F) < \epsilon. Corollary...

Ok, to answer your questions: 1. A movement up or down along the x-value. 2. I'm not sure, we can get the supremum of all finite sums of something by taking the l-inf norm of the sum of all of them.

So, we need to use norms? If we add the assumption of completeness, we will satisfy bounded variation. Since we have Thm: BV[a,b] is complete under llfllBV = lf(a)l + vbaf and Lemma: llfllinf \leq llfllBV then we can prove that llfllBV is complete and therefore the supremum of f is...

I can't get the cartesian product off of there, so please just ignore it.

1. Homework Statement [/b] If f has a continuous derivative on [a,b], and if P is any partition of [a,b], show that V(f,P)\leq \intablf'(t)l dt. Hence, Vba\leq\intablf'(t)ldt. Homework Equations Monotone function \subset BV[a,b] \sumf(ti+1)-f(ti) = lf(b) - f(a)l The Attempt at a...

Homework Statement Given a sequence of scalars (cn) and a sequence of distinct points (xn) in (a, b), define f(x) = cn if x = xn for some n, and f(x) = 0 otherwise. Under what condition(s) is f of bounded variation on [a,b]? Homework Equations Vbaf = supp(\Sigmalf(ti) - f(ti-1)l< +inf...

Homework Statement Prove that f:(M,d) -> (N,p) is uniformly continuous if and only if p(f(xn), f(yn)) -> 0 for any pair of sequences (xn) and (yn) in M satisfying d(xn, yn) -> 0. Homework Equations The Attempt at a Solution First, let f:(M,d)->(N,p) be uniformly continuous...

Really? Are you sure? What we are showing matches my definition of totally bounded nearly verbatim.

Homework Statement Let (X,d), (Y,p) be metric spaces, and let f,fn: X -> Y with fn->f uniformly on X. Show that D(f)c the union of D(fn) from n=1 to n=infinity, where D(f) is the set of discontinuities of f. Homework Equations The Attempt at a Solution Ok, so this looks pretty...

Oh, ok, thanks I appreciate your help.

Alright, lfn(xn) - f(x)l <= lfn(xn) - fn(x)l + lfn(x) - f(x)l <= lfn(xn) - fn(x)l + llfn - fll inf -> lfn(xn) - fn(x)) + 0 as n->inf. So, now we need to show that lfn(xn) - fn(x)l -> 0. Can we do this the same way? lfn(xn) - fn(x)l <= lfn(xn) - f(xn)l + lf(xn) - fn(x)l -> lf(xn) -...

Ok, thanks. I really appreciate it.

Homework Statement Let (X,d) and (Y,p) be metric spaces, and let f, fn: X -> Y with fn -> f uniformly on X. If each fn is continuous at xcX, and if xn -> x in X, prove that lim n-> infinity fn(xn) = f(x). Homework Equations llxll inf (the infinity norm of x) = max (lx1l,...,lxnl)...

Ok, ok, I'm understanding. Fix E>0 and let xc[x-1,x+1]. Since fn -> f uniformly on [x-1,x+1], for some n, p(fn(z), f(z)) < E/3 for all zc[x-1,x+1]. This fn is continuous at x, so for some delta>0, d(x,y)<delta implies p(fn(x),fn(y))<E/3. Then for such y, d(x,y)<delta, p(f(x)...