Constructing an Increasing Sequence to Prove a Limit is Positive Infinity

In summary, the conversation is about proving the existence of an increasing sequence {x_n} in the interval (a,b) given that the limit of f(x) as x approaches b from the left is positive infinity. The main challenge is that f may not be continuous and there are many possibilities to consider. The key is to determine the definition of lim(x->b-)=+inf to guide the proof.
  • #1
Icebreaker
Suppose f:[a,b)->R is such that lim(x->b-)=+inf. Prove that there exists an increasing sequence {x_n} in (a,b) such that f(x_n)>n for all n.

I don't know where to start. It would be easy if I can prove that f is strictly increasing after some point. f might not be continuous so I can't simply look for all f(x) in the form x^2 or something... Too many possibilities. Any pointers will be helpful!
 
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  • #2
What do you have for a definition of lim(x->b-)=+inf ? This will tell you where to start.
 

What is an increasing sequence?

An increasing sequence is a sequence of numbers where each number is greater than the previous number. This means that the sequence is getting larger as it progresses.

How can you identify an increasing sequence?

An increasing sequence can be identified by looking at the numbers in the sequence and determining if each subsequent number is larger than the previous one. Another way to identify an increasing sequence is to plot the numbers on a graph and see if the line is moving upwards.

What are some examples of increasing sequences?

Examples of increasing sequences include: 1, 2, 3, 4, 5 and 10, 20, 30, 40, 50. Both of these sequences are increasing because each number is greater than the previous one.

How can an increasing sequence be useful in real life?

An increasing sequence can be useful in many ways. For example, it can be used to track the growth of a population or the increase in temperature over time. It can also be used in financial analysis to track the growth of investments or profits.

What are some strategies for creating an increasing sequence?

One strategy for creating an increasing sequence is to start with a small number and add a fixed amount to it each time to get the next number in the sequence. Another strategy is to use a mathematical formula, such as multiplying each number by a certain factor, to generate the sequence.

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