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5. Prove that --> 2(n+1 - 1) for all n e Zt. 6. Prove that n < 2" for all n e Z.
Let ne Nj. Prove that n < 2(6(n)).
6. Prove by induction that forn > 1 72 a2 0 0 b
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212 24" 4. Find a formula for II (1-5) and then p 4. Find a formula fo and then prove it by induction for all integers n > 2.
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e bounded on a nondegenerate interval [a, b]. Prove that f is e 0 there is a partition Pe of 5.1.10. Let f b integrable on [a, b] if and only if given la, b] such that P Po implies IU (f, P)-L(f, P)I < ε.
6. (20 pts) Prove that there is no homeomorphism of the closed interval (-1,1] of the real line onto the closed disc < 1 of the complex plane.
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6. The force constant of the CO group in a peptide link is approximately 1.2 kN m'. At what wavenumber would you expect it to absorb? Treat the CO subunit as a separate molecule to find the reduced mass.
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Q 5 1) Prove that (D-o)nnihi ates Xx for K4n-
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Prove: If lim f(x) = L and lim g(x) = M, then lim(f(x) · g(x)) = L.M. xa xa xa 5. State the converse of #4 above. Next, find a counterexample to the converse of #4 above.
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Prove: If lim f(x) = L andlim g(x) = M, then lim(f(x) + g(x)) = L+M. x-a xna xa 3. State the converse of #2 above. Next, find a counterexample to the converse of #2 above.