### Proving an Inequality : Let an, n>= 1 be an increasing sequence of positive real numbers. Prove that 1/a1 + 2/(a1 + a2) + .... + n/(a1 +... + an) < 4(1/a1 + 1/a2 + ... + 1/an)

Let an, n>= 1 be an increasing sequence of positive real numbers. Prove that 1/a1 + 2/(a1 + a2) + .... + n/(a1 +... + an) < 4(1/a1 + 1/a2 + ... + 1/an) Can anything be said about the constant 4? ---