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Let f / R -> R be a strictly increasing continuous function. If \{a_{n}\} is a sequence in [0, 1] then
the sequence \{f(a_{a})\} is
(b) Bounded
(a) Increasing
(c) Convergent
(d) Not necessarily bounded?
the sequence \{f(a_{a})\} is
(b) Bounded
(a) Increasing
(c) Convergent
(d) Not necessarily bounded?
Most Upvoted Answer
Let f / R -> R be a strictly increasing continuous function. If \{a_{n...
Understanding the Sequence {f(a_n)}
The sequence {a_n} is defined within the interval [0, 1]. Given that f is a strictly increasing continuous function, we can analyze the properties of the sequence {f(a_n)}.
1. Boundedness
- The values of a_n are confined to the interval [0, 1].
- Since f is continuous and strictly increasing, the outputs f(a_n) will lie within the interval [f(0), f(1)].
- Thus, {f(a_n)} is bounded as f(0) and f(1) are finite values.
2. Monotonicity
- If the sequence {a_n} is strictly increasing, then f(a_n) will also be strictly increasing, due to the property of f being strictly increasing.
- However, if {a_n} is not strictly increasing, the sequence {f(a_n)} may not be increasing.
- Therefore, we cannot generalize that {f(a_n)} is increasing without knowing more about {a_n}.
3. Convergence
- A bounded sequence does not necessarily converge unless it is also monotonic.
- If {a_n} converges to a limit L in [0, 1], then f(a_n) converges to f(L) due to the continuity of f.
- If {a_n} does not converge or is not monotonic, {f(a_n)} may not converge.
Conclusion
- The sequence {f(a_n)} is bounded.
- It is not necessarily increasing unless {a_n} is strictly increasing.
- It is not necessarily convergent unless additional conditions on {a_n} are satisfied.
- Therefore, the correct answer is: (d) Not necessarily bounded.
The sequence {a_n} is defined within the interval [0, 1]. Given that f is a strictly increasing continuous function, we can analyze the properties of the sequence {f(a_n)}.
1. Boundedness
- The values of a_n are confined to the interval [0, 1].
- Since f is continuous and strictly increasing, the outputs f(a_n) will lie within the interval [f(0), f(1)].
- Thus, {f(a_n)} is bounded as f(0) and f(1) are finite values.
2. Monotonicity
- If the sequence {a_n} is strictly increasing, then f(a_n) will also be strictly increasing, due to the property of f being strictly increasing.
- However, if {a_n} is not strictly increasing, the sequence {f(a_n)} may not be increasing.
- Therefore, we cannot generalize that {f(a_n)} is increasing without knowing more about {a_n}.
3. Convergence
- A bounded sequence does not necessarily converge unless it is also monotonic.
- If {a_n} converges to a limit L in [0, 1], then f(a_n) converges to f(L) due to the continuity of f.
- If {a_n} does not converge or is not monotonic, {f(a_n)} may not converge.
Conclusion
- The sequence {f(a_n)} is bounded.
- It is not necessarily increasing unless {a_n} is strictly increasing.
- It is not necessarily convergent unless additional conditions on {a_n} are satisfied.
- Therefore, the correct answer is: (d) Not necessarily bounded.
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Let f / R -> R be a strictly increasing continuous function. If \{a_{n}\} is a sequence in [0, 1] thenthe sequence \{f(a_{a})\} is(b) Bounded(a) Increasing(c) Convergent(d) Not necessarily bounded?
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Let f / R -> R be a strictly increasing continuous function. If \{a_{n}\} is a sequence in [0, 1] thenthe sequence \{f(a_{a})\} is(b) Bounded(a) Increasing(c) Convergent(d) Not necessarily bounded? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let f / R -> R be a strictly increasing continuous function. If \{a_{n}\} is a sequence in [0, 1] thenthe sequence \{f(a_{a})\} is(b) Bounded(a) Increasing(c) Convergent(d) Not necessarily bounded? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f / R -> R be a strictly increasing continuous function. If \{a_{n}\} is a sequence in [0, 1] thenthe sequence \{f(a_{a})\} is(b) Bounded(a) Increasing(c) Convergent(d) Not necessarily bounded?.
Let f / R -> R be a strictly increasing continuous function. If \{a_{n}\} is a sequence in [0, 1] thenthe sequence \{f(a_{a})\} is(b) Bounded(a) Increasing(c) Convergent(d) Not necessarily bounded? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let f / R -> R be a strictly increasing continuous function. If \{a_{n}\} is a sequence in [0, 1] thenthe sequence \{f(a_{a})\} is(b) Bounded(a) Increasing(c) Convergent(d) Not necessarily bounded? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f / R -> R be a strictly increasing continuous function. If \{a_{n}\} is a sequence in [0, 1] thenthe sequence \{f(a_{a})\} is(b) Bounded(a) Increasing(c) Convergent(d) Not necessarily bounded?.
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