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MATH 131A section 2: Practice Midterm 2.
Please write clearly, and show your reasoning with mathematical rigor. You may use any
correct rule about the algebra or order structure of
R
from Section 3 without proving it.
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1. Suppose (
s
n
) is a given sequence in
IR
, and suppose limsup(
s
n
)
2
≤
1
.
Show that there is a convergent subsequence.
2. Suppose
b
n
>
0
,n
= 1
,
2
,
3
,...
and
b
n
≤
1
n
2
for
n
≥
100. Show that Σ
b
n
converges.
3.
(a) State the deﬁnition of continuity for a function
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Unformatted text preview: f at a point x ∈ dom ( f ). (b) Show that if f is continuous at (0 , 1) and if lim x → + f ( x ) exists, then there exists a continuous extension ˜ f of f on [0 , 1). 4. (a) State the deﬁnition of diﬀerentiability for a function f at a point x ∈ dom ( f ). (b) Show that f ( x ) =  x  is not diﬀerentiable at x = 0. 1...
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This note was uploaded on 11/15/2010 for the course MATH math 131 taught by Professor Kim during the Spring '10 term at UCLA.
 Spring '10
 Kim
 Math, Algebra

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