# How to prove the following sequence

Prove or disprove the following statements:

1. If the sequence $(a_n)_{n\in \mathbb{N}}$ is bounded/restricted then the sequence $(a_n)_{n\in \mathbb{N}}$ is convergent

2. If the sequence $(a_n)_{n\in \mathbb{N}}$ is convergent then the sequence $(a_n)_{n\in \mathbb{N}}$ is bounded/restricted

I´m not sure how to approach this. I would say that they both are the same. If a sequence has an unique limit than its both bounded and convergent...

• If I guess correctly, your confusion stems from what I would call a bad practice of using the word "if" in definitions, and so if you go through all your definitions and replace the "if" by "if and only if" whenever appropriate (sometimes really "if" is meant!), then you will see what I mean.. Jan 16, 2014 at 12:06
• i translated the questions from a different language..if when as...all possible supplements Jan 16, 2014 at 12:12
• sorry I dont see what you mean. actually i asked for assistance...maybe I was not exactly in my question but I´m not sure how to prove this statements. Your answer is not going that direction at all. but still thx. Jan 16, 2014 at 12:18

The first sentence is not correct. Take $a_n=(-1)^n$. This sequence $(a_n)_{n\in\mathbb{N}}$ is bounded, but it is not convergent.

The second statement is correct. Lets say that the limit is $a$. What does that mean? That all but finitely many members of that sequence are in the interval $(a-1,a+1)$ (for example). Take the biggest (with absolute value) number from the set: those outside $(a-1,a+1)$, $|a-1|$, $|a+1|$ and that is your bound.

The two statements are of the following forms:

(1) If A, then B

(2) If B, then A

They are not equivalent. If you have proven one, it doesn't imply anything about the other.

• this is a very helpfull comment :) Jan 16, 2014 at 12:13
• I don't know what you managed to get so far, but from your question I know that you have an underlying misconception regarding logic, and that is what I want to address first, before we can talk about more specific questions. If you understood the difference between (1) and (2), but cannot understand how to do your original question, then perhaps telling us what you've tried for each one would be good, because they are independent questions. Jan 16, 2014 at 15:22
• these a two different questions are from different sources and with no relation to each other. the topic is sequence. the topic is not logic. if you are not able to constructively contribute then just let it go. Jan 16, 2014 at 16:44
• Well you said "I would say that they both are the same." in your opening question, which suggests that you do not understand the logical difference between them. In general, for any pair of statements in these forms, you need two essentially different methods to prove each of them (whether true or false). Also, if you cannot prove something, you should try finding a counter-example, because in many cases you will either find one or understand better why there are none. Have you tried that? Someone else has provided you with the answers, but I think you are confused with logic, not sequences. Jan 17, 2014 at 9:12