Originally Posted by

**anlys** Can you also show the proof by contradiction?

So, supposed that a is not an infimum of D. So, there must exists a larger lower bound of D, let denote it as b.

i.e. a < b <= x for all x in D

So, there exists a delta > 0 such that (a, a + delta) intersect with D = empty set.

Also, a does not belong to the set D because all elements of D are bounded below by b which is greater than a.

Now, let delta = b-a > 0. This implies that for any sequence {X_n} in D\{a}, |X_n - a| >= b-a

So, this means that {X_n} does not converge to a.

So, we get that for any sequence {X_n} in D\{a}, it is not convergent to a. This implies that a is not a limit point. Contradiction!

Is this correct?