functions – Proving Jensen’s Inequality by NOT using induction.

Let $f:Irightarrow mathbb R$ be a convex function. Show that for any integer $n$, any real numbers $x_1,x_2,ldots, x_nin I$ and any positive real numbers $mu_1, mu_2,ldots, mu_n$ such that $sum_{i=1}^{n}mu_i=1:$
$$sum_{i=1}^{n}mu_if(x_i)geq fbigg(sum_{i=1}^{n}mu_ix_ibigg)$$

Now I specifically want to prove it WITHOUT using induction but I’m stuck at a point. Here’s what I’ve tried:

We know that: $$sum_{i=1}^{n}mu_if(x_i)geq mu_1f(x_1)+f(x_k)sum_{i=2}^{n}mu_ispace text{where $f(x_k)=min{f(x_i)}space forall 1leq ileq n$}$$

Since $sum_{i=1}^{n}mu_i=1$, we would have $sum_{i=2}^{n}mu_i=1-mu_1$

Now since $f$ is a convex function: $$mu_1f(x_1)+f(x_k)sum_{i=2}^{n}mu_i=mu_1x_1+(1-mu_1)f(x_k)geq f(mu_1x_1+(1-mu_1)x_k)$$

Thus: $$sum_{i=1}^{n}mu_if(x_i)geq f(mu_1x_1+x_ksum_{i=2}^{n}mu_i)$$

So if I prove that $f(sum_{i=1}^{n}mu_ix_i)leq f(mu_1x_1+x_ksum_{i=2}^{n}mu_i)$ (if it is true), then I’m done.

But this is the step where I’m stuck. Would you please help me prove this?

THANKS