Decide which of the following are true statements. Provide a short justification for those that are valid and a counterexample for those that are not:
a) Two real numbers satisfy a < b if and only if a < b + l for every l > 0
b) Two real numbers satisfy a < b if a < b + l for every l > 0
c) ) Two real numbers satisfy a b if and only if a < b + l for every l > 0
Solution:
Decide which of the following are true statements. Provide a short justification for those that are valid and a counterexample for those that are not:
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For each of the following statements, either prove it is true, or provide a counterexample to show that it is false. (a) If (sn) is a sequence such that lim sn = 0, then lim inf|sn= 0. (b) If f : [0, 1] + R is a function with f(0) < 0 and f(1) > 0, then there exists CE (0,1) such that f(c) = 0. (c) If I is an interval, f:I + R is continuous on I, and...
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profesor do not accept without explaniation 2. IT/F] Decide if the following statements are true or false. Explain (or give a counterexample for) each answer. a) If f(z) is ontinuous and positive forz > 0 and if linn,f(z) = o, then/fe)drconverges. fdz converges. b) The integral / dr diverges c) If bothf(x)d and g(x)da converge, then (().g())dz also converges. d) For any real number p, the integral dz dive 2. IT/F] Decide if the following statements are true or false....