converting1 said:was looking at a proof of this here: http://gyazo.com/8e35dc1a651cec5948db1ab14df491f8
I have two questions,
why do you set K = max of all the terms of the sequence plus the 1 + |A| term? Why do you need the absolute value of all the terms? i.e. why |a_1| instead of |a_1|?
When a sequence of numbers approaches a specific value as the number of terms increases, it is said to be convergent. This specific value is called the limit of the sequence.
A bounded sequence is one in which all the terms fall within a specific range or are limited in their values. This means that the terms of the sequence do not increase or decrease indefinitely.
Proving that all convergent sequences are bounded is important because it helps us understand the behavior of sequences and their limits. It also allows us to make predictions about the terms of a sequence based on its limit.
To prove that all convergent sequences are bounded, we can use the definition of convergence, which states that for a sequence to be convergent, it must approach a specific value (limit) as the number of terms increases. Using this definition, we can show that the terms of the sequence are limited to a specific range.
No, a sequence cannot be convergent without being bounded. If a sequence is not bounded, it means that its terms are increasing or decreasing indefinitely, which goes against the definition of convergence. A sequence must have a limit (a specific value) for it to be convergent, and this limit is only possible if the terms are bounded.