Question

Given a metric space Y, a point L in Y, and f:[0,infinity) -> Y, f has a limit L element of Y at infinity, written, if for every epsilon greater than 0 there is a C>0 such that if x>C then d(f(x),L) < epsilon. Prove, if f:[0,infinity) -> Y is continuous and has a limit at infinity, then f is uniformly continuous.

Answer #1

Let f be defined on the (0,infinity). Prove that the limit as x
approaches infinity of F(x) =L if and only if the limit as x
approaches 0 from the right of f(1/x) = L. Does this hold if we
replace L with either infinity or negative infinity?

Let (X, d) be a compact metric space and let A ⊆ X. Suppose that
A is not compact. Prove that there exists a continuous function f :
A → R, from (A, d) to (R, d|·|), which is not uniformly
continuous.

Please prove the following theorem:
Suppose (X,p) and (Y,b) are metric spaces, X is compact, and
f:X→Y is continuous.
Then f is uniformly continuous.

Let X, Y be metric spaces, with Y complete. Let S ⊂ X and let f
: S → Y be uniformly continuous. (a) Suppose p ∈ S closure and (pn)
is a sequence in S with pn → p. Show that (f(pn)) converges in y to
some point yp.

Let (X, dX) and (Y, dY ) be metric spaces and let f : X → Y be a
continuous bijection. Prove that if (X, dX) is compact, then f is a
homeomorphism

Given a metric space Z and F⊆X⊆Z define F is relatively
closed in X. Show, F is relatively closed in X if and only if
there is a closed set C⊆Z such that F=C∩X.

Given a metric space Z and F⊆X⊆Z define F is relatively closed
in X. Show, F is relatively closed in X if and only if there is a
closed set C⊆Z such that F=C∩X.

Let E and F be two disjoint closed subsets in metric space
(X,d). Prove that there exist two disjoint open subsets U and V in
(X,d) such that U⊃E and V⊃F

Let (X,d)
be a complete
metric space, and T
a d-contraction
on X,
i.e., T:
X
→ X
and there exists a q∈
(0,1) such that for all x,y
∈ X,
we have d(T(x),T(y))
≤ q∙d(x,y).
Let a
∈ X,
and define a sequence (xn)n∈Nin
X
by
x1 :=
a
and ∀n ∈
N: xn+1
:= T(xn).
Prove, for all n
∈ N,
that d(xn,xn+1)
≤ qn-1∙d(x1,x2).
(Use
the Principle of Mathematical Induction.)
Prove that (xn)n∈N
is a d-Cauchy
sequence in...

Suppose f is continuous for x is greater than or equal to 0,
f'(x) exists for x greater than 0, f(0)=0, f' is monotonically
increasing. For x greater than 0, put g(x) = f(x)/x and prove that
g is monotonically increasing.

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