Question

(a) Prove that the sum of uniformly convergent sequences is also a uniformly convergent sequence.

(b) Prove that if, in addition to part (a), the sequences are bounded, then the product is also uniformly convergent.

Answer #1

Prove that every bounded sequence has a convergent
subsequence.

Given that xn is a sequence of real numbers. If (xn) is a
convergent sequence prove that (xn) is bounded. That is, show that
there exists C > 0 such that |xn| less than or equal to C for
all n in N.

1.- Prove the following:
a.- Apply the definition of convergent sequence, Ratio Test or
Squeeze Theorem to prove that a given sequence converges.
b.- Use the Divergence Criterion for Sub-sequences to prove that
a given sequence does not converge.
Subject: Real Analysis

If (x_n) is a convergent sequence prove that (x_n) is bounded.
That is, show that there exists C>0 such that abs(x_n) is less
than or equal to C for all n in naturals

Prove that X is totally bounded if every sequence of X has a
convergent subsequence. Please directly prove it without using any
theorem on totally boundedness.

(1) State the Bolzano–Weierstraß theorem. Prove that the
following sequences have convergent subsequences: (a) {sin(n)} (b)
2n 2+5n+6 n2+12n+20 cos(n 2 )

Use properties of convergent sequences and the Comparison Lemma
to prove that { [5(−1)n ]/n3 } converges in
R.

If (xn) ∞ to n=1 is a convergent sequence with limn→∞ xn = 0
prove that
lim n→∞ (x1 + x2 + · · · + xn)/ n = 0 .

1) Prove: Any real number is an accumulation point of the set of
rational number.
2) prove: if A ⊆ B and A,B are bounded then supA ≤ supB .
3) Give counterexample: For two sequences {an} and
{bn}, if {anbn} converges then
both sequences are convergent.

If Xn is a cauchy sequence and Yn is also a cauchy sequence,
then prove that Xn+Yn is also a cauchy sequence

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