Let f_{n}= . Find the pointwise limit of fn. Is the convergence uniform?

If I'm not mistaken, f_{n}0, For all x in R.

I believe that for this function the convergence is uniform. By letting N> we have,

< < < .

So our choice of N, only depends on epsilon, for all x in R.

Let f_{n}= . Find the pointwise limit of fn. Is the convergence uniform?

Now f_{n}, for all x in R.

In this case however, there is no uniform convergence because:

So our choice of N would imply taking N > . So our choice of N depends on both epsilon and x. Could it be that for a closed interval of the form [a,b] the convergence is uniform?

Thank you