Confused about the definition of a bounded sequence.

[torquerotates]

Ok so for a sequence x(n) to be bounded it means |x(n)|<=M

but according to by book, if x(n) belongs to some closed interval, say [a,b], x(n) is bounded. That is confusing because say x(n) belonging to [a,b] means a<=x(n)<=b.

How can there exist a M such that -M<=x(n)<=M? this means that x(n) belongs to [-M,M]. If we take [a,b] to be an interval that doesn't contain 0, we get a contradiction.

 

[micromass]

The two are equivalent. Say x(n) is in [a,b]. Take an M such that b<M and -M<a. Then x(n) is in [-M,M].

 

[dexdt]

of course some interval [a,b] may contain -M or M, or if -M < a then it still holds.. et c

 

[torquerotates]

So I guess it also works for an open interval. so if all that we know about x(n) is that x(n) is in (a,b), we can claim that since there exist -M<a and M>b, |x(n)|<=M

 

[HallsofIvy]

If x in in (a, b), take M to be the larger of |a|, |b|. Then -M< x< M.