1. Every Cauchy sequence is a convergent sequence in any matric space, but the converse not true in general. nderlvin g space and
1. Every Cauchy sequence is a convergent sequence in any matric space, but the converse not true in general. nderlvin g space and
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Q1: State whether the following statements are true or false:
1. Every Cauchy sequence is a convergent sequence in any matric space, but the converse is
not true in general.
2. The convergence or divergence of a sequence depends only on the underlying space and not
on the metric.
3. If f:M - S is a continuous function between matric spaces (M, dm) and (S, ds) at a point
pEM, then for every closed set A in S, f-"(A) need not to be closed in M.
4. If f:M - S is a continuous function between matric spaces (M, dm) and (S, ds) at an
isolated point p e M, then lim,-p f(x) = f(P).
S. the sequence () is a Cauchy sequence in ([0, 00). I 1), but it is not convergent.
6. The subspace Q of rational numbers of R, forms a complete subspace of R.
7. Every finite subset of any matric space is complete.
8. If f:R -R be a continuous function on (a, b), then f[a, b] is bounded set but not closed in
R.
9. Let f:M S be a function from a metric spaces (M, dm) and (S, ds). If f is uniformly
continuous on M, then f is continuous on M, but the converse is not true in general.
10. The function f:R -R which is defined as; f(x) = , for all x ER, forms a contraction,
since IS(x) - fU)I<Ix - yl.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97c8e66e-a1ef-4b74-98a0-51f821183b17%2Fd7e4ce3f-6670-4d35-9a27-b00f05670d78%2Fc4pif_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Q1: State whether the following statements are true or false:
1. Every Cauchy sequence is a convergent sequence in any matric space, but the converse is
not true in general.
2. The convergence or divergence of a sequence depends only on the underlying space and not
on the metric.
3. If f:M - S is a continuous function between matric spaces (M, dm) and (S, ds) at a point
pEM, then for every closed set A in S, f-"(A) need not to be closed in M.
4. If f:M - S is a continuous function between matric spaces (M, dm) and (S, ds) at an
isolated point p e M, then lim,-p f(x) = f(P).
S. the sequence () is a Cauchy sequence in ([0, 00). I 1), but it is not convergent.
6. The subspace Q of rational numbers of R, forms a complete subspace of R.
7. Every finite subset of any matric space is complete.
8. If f:R -R be a continuous function on (a, b), then f[a, b] is bounded set but not closed in
R.
9. Let f:M S be a function from a metric spaces (M, dm) and (S, ds). If f is uniformly
continuous on M, then f is continuous on M, but the converse is not true in general.
10. The function f:R -R which is defined as; f(x) = , for all x ER, forms a contraction,
since IS(x) - fU)I<Ix - yl.
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