Could someone show me these?

Show that if lim as n goes to infinity (an) = A, then lim as n goes to infinity of

na1 + (n-1)a2 + ... + 2an-1 + an

----------------------------- = (A/2)

n^2

and

Show that for p in the natural numbers, the lim as n goes to infinity of

(1^p) + (2^p) + ... + (n^p)

------------------------- = 1/(p+1)

n^(p+1)

Thanks in advance for any suggestions or help.