Two monotone sequences proof: Prove lim(an)<=lim(bn)

[Hodgey8806]

Homework Statement

Let (a_n) be an increasing sequence. Let (b_n) be a decreasing sequence and assume that a_n≤b_n for all n. Show that the lim(a_n)≤lim(b_n).

Homework Equations

I will show that my sequence are bounded above and below, respectively. Thereby forcing the monotone sequence to be convergent, they will have to converge to their respective supremum and infimum.

The Attempt at a Solution

Let (a_n) be an increasing sequence.
Let (b_n) be a decreasing sequence. And let a_n≤b_n for all n.
Since a_n≤b_n for n, (b_n) is bounded below by a_n for all n.
Thus since (b_n) is monotonically decreasing, there exists an infimum of (b_n) s.t. lim(b_n)=inf(b_n)≥a_n for all n.
Similarly lim(a_n)=sup(a_n)<b_n for all n.
Since a_n≤sup(a_n)≤inf(b_n)≤b_n for all n,
lim(a_n)≤lim(b_n).
Q.E.D.

 

[LCKurtz]

Homework Statement

Let (a_n) be an increasing sequence. Let (b_n) be a decreasing sequence and assume that a_n≤b_n for all n. Show that the lim(a_n)≤lim(b_n).

Homework Equations

I will show that my sequence are bounded above and below, respectively. Thereby forcing the monotone sequence to be convergent, they will have to converge to their respective supremum and infimum.

The Attempt at a Solution

Let (a_n) be an increasing sequence.
Let (b_n) be a decreasing sequence. And let a_n≤b_n for all n.
Since a_n≤b_n for n, (b_n) is bounded below by a_n for all n.

But you have the bound changing with n. It's easy to fix, but you need to give a single number that is a lower bound for the whole sequence.

Thus since (b_n) is monotonically decreasing, there exists an infimum of (b_n) s.t. lim(b_n)=inf(b_n)≥a_n for all n.
Similarly lim(a_n)=sup(a_n)<b_n for all n.
Since a_n≤sup(a_n)≤inf(b_n)≤b_n for all n,
lim(a_n)≤lim(b_n).
Q.E.D.