This problem was taken from a math book I got after it being recommended in a past course. After multiple days of trying and a dozen pages of notes i could not find a proof. I was able to find a semi-solution on this website (How to prove the inequality by induction), however it uses methods far higher than what is bring used at that point and the solution is not fully complete.
For all natural numbers n, and all non-negative real numbers $a_1,...,a_n$ where $\sum_{i=1}^na_i \leq 1$, prove: $\prod_{i=1}^n(1+a_i) \leq 1+2\sum_{i=1}^na_i$
For context, this is in the earlier chapters of a calculus book (Analysis 1, by Otto Forster, 12th edition). Only induction and the axioms of archimedean ordered fields have been introduced (and easy corollaries), otherwise basic proof techniques and easy set theory have been used (no limits, sequences or even irrationals). The other problems around it were all easy (relevant notes of 2-6 lines), so I'd assume theres something I'm missing. I've looked elsewhere, and I could not find this proof (or even name of the equality) in any way. The book provides no solution or hint.
Blindly trying induction would give an inequality that would only hold iff $\sum_{i=1}^n \leq 1/2$, which leaves the case where both $\sum_{i=1}^{n+1} > 1/2$ and the $a_i$ are relatively close together I tried expanding the product partially and then only applying the induction hypothesis to one term, but I get stuck afterwards or I can still find the easy counterexample of a_1 = a_2 = a_3 = ...
Intuitively, I know that a product is biggest when all factors are equal in size, which would give an opportunity to bound the product by $(1+a/n)^n < exp(a) \leq 3^a \leq 3$ where $a = \sum_{i=1}^na_i$. However, proving $(1+1/n)^n < 3$ is an exercise in itself a few pages later and I don't know how the first fact would be proven easily (without calculus, i.e. what would come several chapters later).
I was able to prove n=0, 1, 2 and 3 manually, but I am struggling to find a way to generalize my approach of ordering the a_i and applying the resulting inequalities, mostly because the left side is very complicated to write in its general form. I also tried to use these smaller cases to reduce the general case, as the right side of the inequality resembles a factor of the left side, but the resulting sum is not the correct one in the end.
Am I on the right track with my approaches or am I missing something obvious? Thanks in advance!