Originally Posted by

**JakeBarnes** I don't entirely understand your question. By definition, $\displaystyle f_n \to f $ pointwise if for each fixed x in the domain, the pointwise limit $\displaystyle \lim_{n \to \infty} f_n(x) = f(x) $. For a pointwise limit, you're considering a sequence of points, $\displaystyle x_1, x_2, x_3, ...$ where the jth point in your sequence is just $\displaystyle f_j(x)$. So for each x and $\displaystyle f_n = \sqrt{x^2 + \frac{1}{n}}$, we have that $\displaystyle f_n(x) \to |x|$ since $\displaystyle \lim_{n \to \infty} \sqrt{x^2 + \frac{1}{n}} = \sqrt{x^2}} = |x| $ (this follows from the square root function being continuous, or a simple epsilon estimate if necessary).

Notice in the definition that the pointwise limit is entirely a local property of a sequence of functions. It does not depend on any other point in the domain. In fact, it's entirely possible for the f_n's to only converge for one point in their domain. By definition, we say that $\displaystyle f_n \to f$ uniformly if $\displaystyle \sup{|f_n(x) - f(x)|} \to 0 $ as $\displaystyle n \to \infty$. From the definition, it's clear that a uniform limit implies a pointwise limit. However, the uniform condition is strictly stronger because it's a global condition on the function. All points in the domain must converge to the limiting function at some uniform rate. It does not make sense to as if a sequence of functions converges uniformly at a point.