MATH 412 ADVANCED CALCULUS HOMEWORK #7
Due November 7, 2007
1. Let X= (xn)⊆Rbe a sequence such that lim(xn) = x, and a∈R. Use the
−K() definition of the limit a sequence to show that lim(xn+a) = x+a.
2. Let X= (xn)⊆Rbe a sequence such that lim(xn) = xand xn≤0 for all n∈N.
Show that x≤0.
3. Let X= (xn)⊆Rbe a sequence. Show that lim(xn) = 0 if and only if lim(x2
n) = 0.
(Hint: Use x2
n=|xn|2)
4. Let X= (xn), Y = (yn)⊆Rbe two seuqences with lim xn= 0. Show that
lim(xnsin(yn)) = 0.
5. Let X= (xn)⊆Rbe a sequence such that lim(xn) = xand x > 0. Show that
there exists K∈Nsuch that xn>0 for all n>K. (Hint: Choose =x
2)
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