We got to prove the generalized Bernoulli's inequality given below for $x_1,...,x_n\geq0$ using induction. $$ \frac{1}{1-\sum_{i=1}^nx_i}\geq\prod_{i=1}^n(1+x_i)\geq 1+\sum_{i=1}^nx_i$$
The right side of this seems pretty simple, and is similar to Generalized Bernoulli's inequality, But the left side doesn't seem to make sense to me. Even when looking for $n=1$ we get for $x\geq0$: $$\frac{1}{1-x} \geq 1+x$$ which is not even defined for $x=1$ and obviously fails for $\forall x>1$. Is there a mistake here or am I miss reading the question?
Edit: He gave us another exercise in the same context, this time: $$ \frac{1}{1+\sum_{i=1}^{n}x_{i}}\geq\prod_{i=1}^{n}\left(1-x_{i}\right)\geq1-\sum_{i=1}^{n}x_{i}$$
Once again the right side is straightforward and easy to prove, but for the left side I am once again stuck because for $n=2$ we get $$\frac{1}{1+x_{1}+x_{2}}\geq\left(1-x_{1}\right)\left(1-x_{2}\right)$$ Which is obviously false for at least $x_1,x_2\geq2$. Am I missing something obvious here?