Math problem-Elementary Analysis

There are several question I would like to as

1.Let{xn}infinity n=1 and {yn}infinity n=1 be two convergent sequences with limn->infinity xn=A and limn->infinity yn=B

Prove lim n-> infinity (xny1+x(n-1)y2+…+x1yn)/n=AB

2.let Sn=(n+1)/n+(-1)^n cos(n*pi/6).Find the set of sub-sequential limits of {Sn}infinity n=1

3.see the attach picture please

4.let f:[a,b]->[a,b] be a continuous function.Prove that f must have a fixed point,(Here, fixed point x0 for f means f(x0)=x0)

5.prove that f(x)=xsin(1/(x^2)) is uniformly continuous on (0,infinity)