Suppose that a(n) goes to a and that a(n) is greater than or equal to b for each n. Prove that a is greater than or equal to b.
I began by supposing for a contradiction that a is less than b.
After this I am not quite sure how to proceed. Any help is appreciated.
Ok so I get
a - b < a(n) - a < b - a which implies that a(n) < b, which is a contradiction to what we are given. Is this correct? Also just a quick question, why do we let epsilon = b - a? In other words, what is the reasoning behind that?
well, the idea is that in order to prove a can't be less than b, we have to show that a < b will force some a(k)< b. but that doesn't tell us "which" k we need to find.
however, we DO know that a(n) converges to a, so, given any postive number, ε, we can always find an N for which a(k) is within ε of a for all k > N.
if a < b, then b-a is positive, so we can use it as our epsilon. so now we have an N for which |a - a(k)| < b - a, for all k > N.
so for all these k, a(k) is closer to a than b is. since b is "above" a, a(k) is "below" b, our desired contradiction.
in other words, we know the a(n) are getting closer to a. if a < b, how close to do we need to be? closer to a than b, that is less than b-a.
this will force a(n) < b, which is what we're trying to show (to get our contradiction).