What's an example of a convergent, yet unbounded sequence?

Question

I'm having trouble of thinking how to do this. I want to use a piecewise function, but I feel like there might be something easier than this.

A convergent sequence is always bounded. http://math.stackexchange.com/questions/213936/prove-convergent-sequences-are-bounded — R_D, Dec 13 '15 at 07:41
Are you talking about a sequence consisting of numbers or a sequence consisting of functions? — Ted Shifrin, Dec 13 '15 at 07:41

score 1 · Answer 1 · answered Dec 13 '15 at 07:52

1

Rise is correct. I can't believe I didn't see this sooner. all convergent sequences are bounded. I was confusing this with convergent functions, which are different. (i.e. $f(x)= \frac {1}{x}$)

answered Dec 13 '15 at 07:52

MC989

269

score -1 · Answer 2 · answered Dec 13 '15 at 07:36

-1

Consider the sequence of functions on the interval $(0,1)$ given by $f_n(x) = nx^n$. Note that $f_n(x)$ converges to $0$ point wise on $(0,1)$. However, $f_n(x)$ is unbounded on $(0,1)$.

answered Dec 13 '15 at 07:36

Adhvaitha

20,657

1

So I prove this then using two subsequences? Doesn't what you gave not work since $n$ implies integers? – MC989 Dec 13 '15 at 07:40
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A sequence is indexed by the natural numbers. – Ted Shifrin Dec 13 '15 at 07:42

What's an example of a convergent, yet unbounded sequence?

2 Answers2