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I had a test in which the question that I will present here was asked. I got no points for my attempt at a solution. Do you think that I was still on the right track and that I deserve partial points? Here is the question:

"A number M is said to be an upper bound to a set A if M [itex]\geq[/itex] x for every x[itex]\in[/itex] A. A number S is said to be supremum of a set A if S is the smallest upper bound to A.

Assume that:

A = {(4n^{2})/(n^{2}+1) : n [itex]\geq[/itex]0 is an integer}.

Show that supremum of A is 4."

And here is what I wrote as an answer (not verbatim, but translated from another language):

"Since n does not have an upper limit, it can go toward infinity. In this case:

A = lim (n [itex]\rightarrow[/itex] [itex]\infty[/itex]) (4n^{2})/(n^{2}+1)=[itex]\infty[/itex]/[itex]\infty[/itex]. This shows that we can use l'hopital's rule. After using l'hopital's rule twice we get that A = 4. In other words, this gives us supremum. Since n always can be even bigger, this is just the smallest upper bound.

Answer: By using l'hopital's rule twice, I have shown that supremum A is 4."

Out of the possible 4 points that one could get on that question, I got 0. Was it justified?

Thanks in advance!

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# Question on Supremum

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