Suppose $\varphi$ is convex on $(a,b)$. I want to show that for any $n$ points $x_1,\dots,x_n \in (a,b)$ and nonnegative numbers $\theta_1,\dots,\theta_n$ such that $\sum_{k=1}^n \theta_k = 1$ we are able to boost $\varphi$'s convexity property up to look like Jensen's Inequality for simple functions:
$$\varphi\left( \sum_{k=1}^n \theta_k x_k \right)\leq\sum_{k=1}^n \theta_k\varphi\left( x_k \right)$$
So far I observed that if $\sum_{k=1}^n \theta_k = 1$ then $(1-\theta_n) = \sum_{k=1}^n \theta_{n-1}$. So, we use this to manipulate the RHS in order to apply $\varphi$'s convexity:
$$\varphi\left(\theta_nx_n + (1-\theta_n)\sum_{k=1}^{n-1} \frac{\theta_k x_k}{1-\theta_n} \right)\leq \theta_n\varphi(x_n) + (1-\theta_n)\varphi\left(\sum_{k=1}^{n-1} \frac{\theta_k x_k}{1-\theta_n}\right)$$
I think induction is waiting for me, but I can't see the next step. Perhaps I am not thinking clearly enough about what the right hand side means in this last line. Any help would be nice.
You are basically done. You just need to apply the inductive hypothesis on your last term.