Math 312 Exam 1 True/False

For any real numbers a and b, |a − b| ≤ |a| − |b|.

False - Let a=-10 and b=6 (16<4 not true)

There exists (sn) such that −5 < sn < 3 for all n and every subsequence of (sn) diverges.

False - Let sn be a bounded subsequence. Then by Bolzano_Weierstrass there must be a convergent subsequence

If sn > a for all n, then lim sn > a

False - consider sn=1/n such that lim sn=0

Every bounded sequence converges.

False - divergent bounded sequences ( sn=n bounded below by 1 but diverges)

The product of two divergent sequences diverges.

False - let an=bn=(-1)^n

Every monotone sequence converges

False - let an=n

If (sn) converges and sn ≠ 0 for all n, then (1/sn) converges

False - let sn=1/n

If lim inf sn ∈ R and lim inf tn ∈ R, then lim inf (sntn) = (lim inf sn) * (lim inf tn)

False - let sn={1,2,1,2,1,2,...} and tn={2,1,2,1,2,1,2...}

If the set of values of a sequence is { 1/n : n ∈ N}, then the sequence converges

False - say sn={-1,-1,-1...} and tn=(0,0,0,0,....)

If (sntn) and (sn) converge, then (tn) also converges

False - sn=1/n & tn=n

Every sequence has an increasing subsequence

False - take sn=1/n

If (sn + tn) converges, then both (sn) and (tn) converge.

False- Let sn = n and tn = −n. Then sn + tn = 0 for all n so (sn + tn) converges, but both (sn) and (tn) diverge.

If s < M for all s ∈ S, then sup S < M. S is a non-empty subset of R.

False- S=(0,1) and so M=1=Sups

If for each ε > 0 there are infinitely many n such that |s − sn| < ε, then (sn) converges to s

False- Sn =(-1)^n then there are infinitely many n at | 1-sn |=0<ε. However, sn diverges with s=1.

If the set of values of a sequence is {−1, 1}, then the sequence diverges

False- {-1,1,1,1,1,1,...} converges to 1

An unbounded sequence must diverge to +∞ or −∞.

True

Every Cauchy sequence converges

True

Every bounded monotone sequence converges

True

The limit of a convergent sequence is unique.

True

Every convergent sequence is a Cauchy sequence

True -

Every Cauchy sequence is bounded

True - All convergent sequences must be bounded and monotonic

If (sn + tn) and (sn) converge, then (tn) also converges

True - If (sn) and (sn+tn) converge then (tn)=((sn+tn)-sn) must also converge

If a ≤ sn ≤ b for all n and (sn) converges, then a ≤ lim sn ≤ b

True - Look at written T/F for answer

If lim sup sn = b, then no subsequence of (sn) converges to a number greater than b.

True - Proof in notes on T/F

Every convergent sequence is bounded.

True - Theorem 29

If b = lim sup sn, then there is a number N such that sn ≤ b for all n > N

True - after negating the statement you get the contrapositive that some a exists such that sn>b

If lim inf sn ≠ lim sup sn, then (sn) diverges

True - contrapositive of the idea that liminf an =limsup an=a such that an converges and lim an = a

The product of two convergent sequences converges.

True - lim an -> a & lim bn -> b then (lim an)(lim bn)= lim(an * bn)

If |sn| ≤ tn for all n and tn → 0, then sn → 0

True- Squeeze Theorem since -tn<sn<tn and tn converges to 0

If m > inf S, then there exists s ∈ S such that s < m.S is a non-empty subset of R.

True- definition of inf

If lim sn = 1, then .9 < sn < 1.1 for all n except for finitely many

True- take ε=0.1. Since lim exists there exists some N at n>N implies | Sn-|<ε , that ε is true for all n larger than N

For any real numbers a and b, |a − b| ≤ |a| + |b|.

True-Triangle Inequality