Expectation of the maximum of n random variables?

Question

Let's say we have $n$ independent random variables, each variable equally likely to take any value in the interval $[0,1]$. What is the expectation of the maximum of these $n$ random variables?

Progress so far :-

I came up with a intuitive sort of solution. Looking for a rigorous one. On an average, we can say that the $n$ points would be equally spaced on the number line, as no variable is special and no point on the number line is special. Hence Expectation = $\frac {n}{n+1}$ as this is the right most point on an average.

Answer 1

HINT

If $X = \max(X_1,\ldots,X_n)$, then $X < c$ iff $X_i < c$ for all $i$. Probabilities multiply over independent events.