How do you show a sequence is monotonically decreasing?

Section 4-2 : More on Sequences

1. We call the sequence increasing if an
2. We call the sequence decreasing if an>an+1 a n > a n + 1 for every n .
3. If {an} is an increasing sequence or {an} is a decreasing sequence we call it monotonic.

How do you prove that a sequence is not bounded?

If a sequence is not bounded, it is an unbounded sequence. For example, the sequence 1/n is bounded above because 1/n≤1 for all positive integers n. It is also bounded below because 1/n≥0 for all positive integers n. Therefore, 1/n is a bounded sequence.

Is a decreasing sequence bounded below?

The sequence (n) is bounded below but is not bounded above because for each value C there exists a number n such that n>C. Figure 2.4: Sequences bounded above, below and both. Each increasing sequence (an) is bounded below by a1. Each decreasing sequence (an) is bounded above by a1.

Are all monotonic sequences bounded?

Only monotonic sequences can be bounded, because bounded sequences must be either increasing or decreasing, and monotonic sequences are sequences that are always increasing or always decreasing. The smallest value of an increasing monotonic sequence will be its first term, where n = 1 n=1 n=1.

Are monotonic sequences bounded?

Only monotonic sequences can technically be called “bounded” Only monotonic sequences can be bounded, because bounded sequences must be either increasing or decreasing, and monotonic sequences are sequences that are always increasing or always decreasing.

How do you prove if a sequence is monotonically increasing?

A sequence (an) is monotonic increasing if an+1≥ an for all n ∈ N. The sequence is strictly monotonic increasing if we have > in the definition. Monotonic decreasing sequences are defined similarly. A bounded monotonic increasing sequence is convergent.

Is a monotonic sequence always bounded?

Only monotonic sequences can be bounded, because bounded sequences must be either increasing or decreasing, and monotonic sequences are sequences that are always increasing or always decreasing.

What is monotonic bounded sequence?

We will learn that monotonic sequences are sequences which constantly increase or constantly decrease. We also learn that a sequence is bounded above if the sequence has a maximum value, and is bounded below if the sequence has a minimum value.

How do you prove something is bounded below?

Here it is. S is called bounded below if there is a number m so that any x ∈ S is bigger than, or equal to m: x ≥ m. The number m is called a lower bound for the set S. Note that if m is a lower bound for S then any smaller number is also a lower bound.

Can a divergent monotone increasing sequence have an upper bound?

But we are given A n is a divergent monotone increasing sequence. Hence A n cannot be bounded above; i.e., A n has no upper bound. (That’s simply applying the contrapositive of the monotone convergence theorem).

Is every term in a monotonically decreasing sequence infinite?

Then, being monotonically decreasing, every term a_n would be infinite, which leads to a contradiction because then a_n us bounded below by every real number. Let the sequence be a n. Then, since a n is not bounded below, it means that given any N, ∃ n 0 such that a n 0 < N. But since a n is decreasing, a n < a n 0 ∀ n > n 0.

What is the contradiction of a_N diverging to +infinity?

Suppose a_n divergs to +infinity. Then, being monotonically decreasing, every term a_n would be infinite, which leads to a contradiction because then a_n us bounded below by every real number. Let the sequence be a n.

What is the limit of the decreasing sequence x n?

Hencee, indeed, the sequence ( x n) is decreasing and bounded below by 0 . So it converges to some limit L ≥ 0.