What's an example of a convergent, yet unbounded sequence?
By Emma Terry •
What's an example of a convergent, yet unbounded sequence?
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.
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$\begingroup$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. $f(x)= \frac {1}{x}$)
$\endgroup$ $\begingroup$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 $(0,1)$.
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