OME 1973 Problem 3

The sequence $(a_n)$ of complex numbers is considered in the complex plane, in which is: $$a_0 = 1, \,\, a_n = a_{n-1} +\frac{1}{n}(\cos 45^o + i \sin 45^o )^n.$$ Prove that the sequence of the real parts of the terms of $(a_n)$ is convergent and its limit is a number between $0.85$ and $1.15$ .