Median-of-Medians proof for time complexity
Assuming that you got the base covered, here is the inductive step:
$$
T(n) \leq T(n/5) + T(7n/10) + cn \leq 10c(n/5) + 10c(7n/10) + cn = 10cn.
$$
(Cheating a bit, since $n$ isn't necessarily divisible by 10; but that's a problem in the recurrence.)

You can also just apply the Akra–Bazzi theorem.