A sequence of functions fn:S→ℝ on a given set S converges pointwise if the real sequence fn(x) converges for all x∈S. This means for all x in S and for every ε > 0, there is an integer N such that for all m,n ≥ N |fm(x)−fn(x)| < ε. Note that N may depend on how both x and ε are defined. This is in contrast to uniform convergence, where, for any given ε, a single N has to exist for all values of x.