Expected Number of Uniforms to Exceed a Threshold

Draw i.i.d. uniform random variables $X_1, X_2, \ldots$ from $U[0,1]$ and let $S_n = X_1 + X_2 + \cdots + X_n$. Define the stopping time $T = \min\{n : S_n > 1\}.$ 1. Show that $E[T] = e$. 2. Generalize: for a threshold $c > 0$, let $T_c = \min\{n : S_n > c\}$. Find $E[T_c]$.