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Unformatted text preview: MATH 1313 (Winter 2003, Lecture 1)
Instructor: Roberto Schonmann
Midterm Exam Last Name:
First and Middle Names:
Signature: UCLA id number (if you are an extension student, say so): Provide the information asked above and write your name on the top of
each page using a pen. You should show your work and explain what you are
doing; this is more important than just ﬁnding the right answer. You can use
the blank pages as scratch paper or if you need space to ﬁnish the solution
to a question. Please, make clear what your solution and answer to each
problem is. When you continue on another page indicate this clearly. You
are not allowed to sit next to students With Whom you have been studying
for this exam or to your friends. Good Luck ! Score 1) (10 points) Let {(xn,yn)} be a sequence of points in a rectangle S =
[a, b] X [0, d]. Prove that {(xn,yn)} has a subsequence that converges to a
point of S. [Hintz use the corollary to the BolzanoWeirstrass theorem which
states that any sequence {Zn} of real numbers in a ﬁnite closed interval [7", s] has a subsequence that converges to a point of [7", W M k aw 3 {xmkg Age. ’14“ —> 7:. , 5% me («767. CD CM, ) “'5 0st 6. Calailegou, ﬁ 2) (10 points) Suppose that f and g are @discontinuous functions from [R to
IR, and that {fn} is a sequence of functions from 1R to R such that fn ——> f
pointwise. Can you conclude that g fn —+ gf pointwise? Prove your answer.
[Nokcredit will be given for a right answer given for the wrong reason] HM 8C7») QC») ——> £093 ’v‘zetﬁ.
Una," a... Wm; 4:4, I : am—sa. m) mommat). 3) (10 points) Prove the theorem that states that if {fn} is a sequence of
continuous functions on [a, b] and fn —> f uniformly, then 4" fn(m)dac ‘—> b mm. [Recall that this is a theorem proved in Section 5.2.] gawk a,.>¢ 3N 3,75, MN a...) {wagmls 3» was? L L b lav*4.»
=5? gﬁmmlz “Q€W>Jw\g \St£wtw3—%Lm>)&w\ N
p
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3:
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E3 4) (10 points) For f E C[a, b], deﬁne Hle = fff(x)da:. Show that  “1
satisﬁes the triangle inequality v f+gl1 S f1+ llgll1 5) (10 points) Suppose that (M, p) is a metric space with the property that
for all 117,3; 6 M, p(a:,y) E {0, 1, 2,3, Show that if is a sequence of
points in M which converges to a point a: E M, then there exists N such that
nZN => mnzx. T0)“ 8’3‘ y“). . xvi427;.) 96,
M; N m> P(%)9L3 as: say:
Q)
:2; Pam») ..: 0 CD What. é {DHJZNM} . ...
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