बोलजनों-वेयरस्ट्रास प्रमेय लिखिए। इसका प्रयोग करके जाँच कीजिए कि समुच्चय का कोई सीमा-बिंदु है या नहीं।
बोलजनों–वेयरस्ट्रास प्रमेय के अनुसार वास्तविक संख्याओं के किसी भी अनन्त तथा परिबद्ध समुच्चय में कम-से-कम एक सीमा-बिंदु अवश्य होता है। दूसरे शब्दों में, यदि (E \subset \mathbb{R}) अनन्त है और किसी वास्तविक संख्या (M) के लिए ___ __________ ________ ___ ______ _________.
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सिद्ध कीजिए कि यह समुच्चय संवृत नहीं है।
Show that the function defined as
is continuous on its domain, but not uniformly continuous there. Will the function be uniformly continuous on
for any fixed c > 0?
State whether the following statements are true or false. Justify your answers with a short proof or a counterexample.
i) The negation of is
.
ii) is countable.
iii) The equation has a real root between -2 and 1.
iv) Every increasing sequence has a convergent subsequence.
v) The function defined as
has a removable discontinuity.
vi) An integrable function can have finitely many points of discontinuity.
vii) The series is divergent.
viii) The function f defined by , has local maximum at
.
ix) Every constant function on [a, b] is Riemann Integrable.
x) The function defined by
is differentiable at
.
Examine the type of discontinuity of the function at where
Give one example for each of the following. Justify.
(i) A set which is neither open or closed.
(ii) A set which has no limit point.
यदि अनंत श्रेणी अभिसारी है, तो दिखाइए कि
होती है।
गणितीय आगमन का प्रयोग करके सिद्ध कीजिए कि प्रथम 'n' प्राकृतिक संख्याओं के वर्गों का योगफल होता है।