Does Uniform Convergence Imply Pointwise?

Uniform convergence implies pointwise convergence, but not the other way around. For example, the sequence fn(x)=xn from the previous example converges pointwise on the interval [0,1], but it does not converge uniformly on this interval.

Why does uniform convergence imply pointwise?

In uniform convergence, one is given ε>0 and must find a single N that works for that particular ε but also simultaneously (uniformly) for all x∈S. Clearly uniform convergence implies pointwise convergence as an N which works uniformly for all x, works for each individual x also. However the reverse is not true.

Does uniform convergence imply limit?

It turns out that the uniform convergence property implies that the limit function f inherits some of the basic properties of { f n } n = 1 ∞ \{f_n\}_{n=1}^{\infty} {fn}n=1∞, such as continuity, boundedness and Riemann integrability, in contrast to some examples of the limit function of pointwise convergence.