Question

show that a sequence of measurable functions (f_{n})
converges in measure if and only if every subsequence of
(f_{n}) has subsequence that converges in measure

Answer #1

(fn) is a sequence of entire functions, which converges
uniformly to an entire function f on every compact subset K of C,
and f is not identically zero. prove that if fn only have real
roots, then f only has real roots too.

Show that sequence {sn} converges if it is monotone
and has a convergent subsequence.

Suppose (an) is an increasing sequence of real numbers. Show, if
(an) has a bounded subsequence, then (an) converges; and (an)
diverges to infinity if and only if (an) has an unbounded
subsequence.

Show there does not exist a sequence of continuous functions fn
: [0,1] → R converging pointwise to the function f : [0,1] → R
given by f(x) = 0 for x rational, f(x) = 1 for x irrational.

construct an example of sequence of functions that the family of
such function is uniformly bounded but does not have subsequence
that converges uniformly, with detail proof.

Show that if sequence (an) converges, then all the rearrangement
of (an) converges, and converge to the same limit

Suppose that every Cauchy sequence of X has a convergent
subsequence in X. Show that X is complete.

Let Fn denote the nth term of the Fibonacci Sequence.
Show that Fn is less than or equal to 2(n-1)
for all natural numbers n through mathematical induction.

Problem 3. Consider a sequence for functions
fn : [0, 2] → R such that f(0) = 0 and
f(x) = (sin xn)/(xn) for x ∈ (0, 2]. Find
limn→∞ ∫[0,2] fn(x) dx

Complex Analysis
Give a sequence of real functions differentiable on an interval
which converges uniformly to a nondifferentiable function.

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