Prove that the linear combination of points in a convex set must be in the set

Question

Let K be a convex set. Suppose $x_1,x_2,...,x_n \in$ K. Prove that: $$ x=\sum_1^n a_jx_j \in K, \space\space\space\space\space\space\space where \space\space\space\space\space\space\space\sum_1^na_j=1 $$ I tried to prove this by induction but failed because I couldn't handle the condition $\sum_1^na_j=1$ for the $(n+1)^{th}$ case.

Answer 1

$\sum_1 ^{n+1} a_j x_j =\alpha \sum_1 ^{n} b_j x_j +(1-\alpha) x_{n+1}$ where $\alpha =\sum_1 ^{n} a_j$, and $b_j = \frac {a_j} { \sum_1 ^{n} a_j}$. First conclude that $\sum_1 ^{n} b_j x_j$ is in K and then use definition of convexity again to finish the proof. ( In writing above identity I have used the fact that $1-\alpha =1-\sum_1 ^{n} a_j =a_{n+1}$.