I was reading a proof of Jensen's inequality on convex functions, and I need some help understanding it. The proof is as follows:
$$f(t_1x_1+\dots+t_nx_n) = f((1-t_n)\left(\frac{t_1}{1-t_n}x_1+\dots+\frac{t_{n-1}}{1-t_n}x_{n-1}\right)+t_nx_n)$$ $$\le (1-t_n) f\left(\frac{t_1}{1-t_n}x_1+\dots+\frac{t_{n-1}}{1-t_n}x_{n-1}\right)+t_nf(x_n),(\text{convexity})$$ $$\le (1-t_n)\left[\frac{t_1}{1-t_n}f(x_1)+\dots+\frac{t_{n-1}}{1-t_n}f(x_{n-1})\right]+t_nf(x_n),(\text{induction})$$ $$=t_1f(x_1)+\dots+t_nf(x_n).$$
The proof is fairly simple but can someone please explain the inductive step for me? Thanks.
The inductive assumption is that $$f(\lambda_1 x_1+\dots+\lambda_{n-1}x_{n-1})\leq \lambda_1 f(x_1)+\dots+\lambda_{n-1}f(x_{n-1})$$ already holds if $\sum_{j=1}^{n-1}\lambda_j=1$ and since $\sum_{j=1}^{n}t_j=1$ one has
$\sum_{j=1}^{n-1}\frac{t_j}{1-t_n}=\frac{1-t_n}{1-t_n}=1$.