00:02
We want to find this supremum and infimum of the set a equal 1 plus n over 2n minus 1 over n for n a natural number.
00:14
So first, a is not an m to set and that is evident because 1 is a natural number implies that 1 plus 1 over 2 times 1 minus 1 over 1 belongs to a.
00:38
That is, 0 belongs to a.
00:45
And that is because for n -equal 1, we get, putting n -equal 1 in the expression defining the elements of a, we get the number 0.
00:56
So 0 is an element of a.
01:00
In fact, there are infinitely many numbers in a because we can put as n any natural number.
01:09
But sufficient to give one example of an element of a to say that a is not the empty set.
01:17
So the second thing is the expression defining the elements of a can be simplified in some way.
01:27
So a is equal to 1 plus n over 2n.
01:36
Or let's say instead of doing this way, i'm going to simplify the expression for n, a natural number we have that 1 plus n over 2n minus 1 over n is the same as 2n as a common denominator we get 1 plus n minus 2 let's say i write this 2 properly minus 2 and that is n minus 1 .m 2 and that is n minus 1 over 2 n so a is a is is exactly equal to the set of numbers of the form n -n minus 1 over 2n for n natural number.
02:32
It's exactly the same because the expression defining the elements of a is exactly equal to this other expression.
02:41
So instead of using this given expression, we can use this other here.
02:50
So we have that.
02:52
And then we're going to prove that or verify first that all the elements in a are positive and that's obvious because we have 2n in the denominator for a natural number n2 times n is positive because n is positive and n minus 1 is positive or 0.
03:18
So the only element that is 0 in a is corresponds to n equal 1.
03:26
Any other number is positive.
03:28
So we have zero and positive numbers so we can say that all the elements in a are non -negative.
03:53
That is they are positive or zero.
03:59
That's evident for what we said before that is n minus 1 can be 0 but in it for n equal 1 but in any of the case that's positive.
04:11
Because n becomes being 2, 3, 4, et cetera, and for that n minus 1 always positive.
04:19
And the denominator is always positive for n in the natural number.
04:24
So we have a non -negative number for all the elements of 8.
04:29
That is, this means, a is bounded below by 0.
04:43
So that's the first thing we can say.
04:46
Is zero and let's see if we if it is founded above what we're going to see is that it is bounded above and we're going to prove that this set is bounded above by one half so a is above by one half let's see that we get to show that n minus 1 over 2n is less than or equal to one half and that's equivalent to the expression n minus 1 less than are equal to n and that's equivalent to the expression negative 1 less than equal to 0 which is true in fact negative 1 is less than 0 but this is true we can add n both sides we get this we can write n as 2n over 2 and then multiply by 1 over 2 n both sides and we get this.
06:19
So these are equivalent inequalities for n in the natural numbers.
06:29
We have that for any natural number n.
06:41
So one half is an upper bound for a.
06:55
And from the previous observation we can say that a has both supremum and infimum...