Okay, so I have a problem where and I need to show using the formal definition that converges to 0:
So I let . Simplifying it, I get
, essentially getting within distance of for any integer where .
So then I get .
Doing some manipulation, I eventually get to This is where I get lost on showing that .
Now my professor didn't really sufficiently explain how to do a formal proof even when we already assume we know the value of . For instance, we know for this . Knowing this, the inequality would become . Do we know that this is true because is always less than since , or is there more that needs to be shown? If that is sufficient, than how can that line of thought be applied when we assume we don't know the value of (like in this problem)?
Sorry if there is already a thread that asks and explains this. Our class pretty much went through sequences today in about 30 minutes so I'm not too sure on this stuff. Thanks!
Hmm, so correct me if I'm wrong, so would that imply then that:
Since , cannot be positive or the inequality does not hold true. But how exactly can we show is never negative, and therefore .
Also, doesn't the information you provided already assume ? Sorry if these are not the best of questions, but like many others, my teachers and professors have always neglected the formal proofs and definitions (which sucks because I plan on being a pure math major). Thanks again!