[b]1. Let r(n) = (1+1/n)^n and t(n) = (1+1/n)^n+1. (Use r(n) converge to e).
Show that t(n) > r(n) for all n and that lim n->inf(t(n) - r(n)) = 0.
Show that {tn} is a decreasing sequence with limit e. {Hint: express {(1+1/n-1)/(1+1/n)}^n as (1+a)^n and apply Bernoulli's inequality). Use n=10...
Let A, B in R^n be closed sets. Does A+B = {x+y| x in A and y in B} have to be closed?
Here is what I've tried. Let x be in A^c and y in B^c which are both open since A & B are closed. So for each x in A^c there exists epsilon(a)>0 s.t. x in D(x, epsilon(a) is subset of A^c. For each y...