Let Sn(x)= nx/(nx+1)

Compute the pointwise limit of Sn on [0,1].

Is the convergence uniform?

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- Feb 20th 2008, 05:51 PMcowgirl123pointwise limit?
Let Sn(x)= nx/(nx+1)

Compute the pointwise limit of Sn on [0,1].

Is the convergence uniform? - Feb 20th 2008, 06:15 PMJhevon
$\displaystyle S_n \to S$ point-wise on [0,1] where $\displaystyle S = \left \{ \begin{array}{lr} 0 & \mbox{ if } x = 0 \\ & \\ 1 & \mbox{ if } x \in (0,1] \end{array} \right.$

all we did was to take the limit as $\displaystyle n \to \infty$

Is it uniform? there are several ways to attack this.

what do we know about the limit of a uniformly continuous function?

we could also use the $\displaystyle \epsilon - \delta$ definition for uniform convergence

we could also check to see whether or not $\displaystyle \lim_{n \to \infty} [ \sup \{ |S(x) - S_n(x)|~:~x \in [0,1] \}] = 0$ in which case it is uniformly convergent if it is 0