Question
State Bolzano-Weierstrass theorem. Use it to check whether the set:
has a limit point.
Answer :
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Bolzano–Weierstrass theorem (sequence form): every bounded sequence in (\mathbb{R}^n) has a convergent subsequence. Equivalently for sets in (\mathbb{R}): every infinite bounded subset of (\mathbb{R}) has at least one accumulation (limit) point. Consider the set [ S=\left{\frac1m-\frac1n:;m,n\in\mathbb{N}\right}. ] First note (S) is bounded: for all (m,n) we have (-1<\frac1m-\frac1n<1), so (S\subset(-1,1)). Also (S) is infinite (there are infinitely many ordered pairs ((m,n)) producing distinct values), so by Bolzano–Weierstrass (S) must have at least one accumulation point. We now identify explicit limit points.
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Related Questions
If an infinite series converges, then show that
.
Suppose that f and g are in R[a, b]. Then show that the function f + g is in R[a, b], and
जाँच कीजिए कि अनुक्रम , जिसे इस प्रकार से परिभाषित किया जाता है:
, और
.
अभिसारी है या नहीं।
Check whether the following sets are open or closed or neither:
(i)
(ii)
Check whether or not the sequence , defined by:
and is convergent.
निम्नलिखित में से प्रत्येक के लिए एक - एक उदाहरण दीजिए। अपने उत्तर का कारण बताइए।
i. एक ऐसा समुच्चय जो न तो विवृत है और न संवृत है।
ii. एक ऐसा समुच्चय जिसका कोई सीमा बिंदु नहीं है।
Show that the set:
is not closed.