(1) Let [0,1]* be the set of all functions N-->[0,1] (ie of all sequences in [0,1]). Define a sequence (Fn) in [0,1]* by:

Fn(k)= 0 if k different from n; 1 otherwise.

Consider the following metric on [0,1]*: d1(f,g):=sup{|f(n)-g(n)|:n element of N}.

Determine whether (Fn) converges with respect to d1.

(2) same as in (1) but with respect to the metric d2(f,g):= (sum from i=0 to infinity)[(2^-i)|f(i)-g(i)|.

Note that N is the set of all natural numbers including zero.

I do not know how to check convergence with respect to a metric.