Prove that if f is a bounded function on a bounded interval
[a,b] and f is continuous except at finitely many points in [a,b],
then f is integrable on [a,b]. Hint: Use interval additivity, and
an induction argument on the number of discontinuities.
(a) Prove that Sn is generated by the elements in the set {(i
i+1) : 1≤i≤n}.
[Hint: Consider conjugates, for example (2 3) (1 2) (2
3)−1.]
(b) ProvethatSn isgeneratedbythetwoelements(12)and(123...n) for
n ≥ 3.
(c) Prove that H = 〈(1 3), (1 2 3 4)〉 is a proper subgroup of
S4.
1: (an) and (bn) are bounded
sequences:
(a) prove that limsup(-an) =
-liminf(an)
(b) for any c>0, prove that
limsup(can) = climsup(an)
and
liminf(can) = climinf(an)
(c) prove that
limsup(an+bn) ≤ (limsup(an)) +
(limsup(bn))
and
liminf(an+bn) ≥ (liminf(an)) +
(liminf(bn))
(d) If an and bn are made of nonnegative
terms, prove that
limsup(anbn) ≤ (limsup(an)) x
(limsup(bn))
and
liminf(anbn) ≥ (liminnf(an)) x
(liminf(bn))
(e) prove that
limsup(an+1) = limsup(an)
and
liminf(an+1) = liminf(an)