(Solved): Problem 4(15+15=30 points ) (a) Consider the sequence (fn) defined as follows: fn(x)={x/n1/n ...

Problem $$4(15+15=30$$ points $$)$$ (a) Consider the sequence $$\left(f_n\right)$$ defined as follows: $$f_n(x)=\{\begin{array}{ll}x/n& \text{ if }n\text{ is even }\\ 1/n& \text{ if }n\text{ is odd }\end{array}$$ First, determine the pointwise limit. Next, show that for every $$a>0$$ the sequence $$\left(f_n\right)$$ converges uniformly on the interval $$[?a,a]$$. Is the convergence uniform on $$\mathbb{R}$$ ? (b) Consider the sequence $$\left(g_n\right)$$ defined as follows: $$g_n(x)=\cos (x/n).$$ First, determine the pointwise limit. Next, show that for every $$a>0$$ the sequence $$\left(g_n\right)$$ converges uniformly on the interval $$[?a,a]$$. Is the convergence uniform on $$\mathbb{R}$$ ?