According to wikipedia:
Jensen's inequality is
${\displaystyle f(tx_{1}+(1-t)x_{2})\leq tf(x_{1})+(1-t)f(x_{2}).}$
With this, I can easily prove Inequality of arithmetic and geometric means, in other words to prove that:
${\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}\geq {\sqrt[ {n}]{x_{1}\cdot x_{2}\cdots x_{n}}}\,,$
My problem is that my teacher didn't define Jensen's inequality in the same way above but rather:
Let $X$ be a discrete random variable with finite expected value and let $h:\mathbb{R} \to \mathbb{R}$ be a convex function. then: $h(E[X])\leq E[h(X)]$
How can I reach (prove) the first version using my professor's version of the inequality, so I can prove the Inequality of arithmetic and geometric means?
The way your prof stated Jensen's inequality is more general. The inequality $$f(tx_1+(1-t)x_2)\leq tf(x_1)+(1-t)f(x_2),\quad t\in[0,1]\quad [1]$$ is a special case of
$$f(E[X])\leq E[f(X)]\quad \text{Jensen's inequality}$$
for $f$ convex, where the random variable $X$ has two-point support $\{x_1,x_2\}$ with respective probability masses $t,1-t.$ However, it is important to note that $[1]$ is actually the definition of $f$ being a convex function, so one doesn't require Jensen's inequality to justify it.