I just want to add a simple piece to this reasoning, that is very intuitive and not excessively mathematic, since the mathematic explanation has already been given in the other answers (I like to base my mathematical understanding on logic intuitive reasoning).
Just consider what a put option is: it is a contract to sell at the strike K and buy at the final price P of the underlying. Consider your percentage return on a put, which is K/P-1. When P is close to 0 but it is not yet 0, your percentage return is high but still discrete because at maturity you will have to pay P. Assume now that P is exactly 0 (which in practice means that the company will never recover and your contract will be paying for sure at maturity T, or maybe will be terminated even sooner if in real practice that are clauses in certificates allowing to terminate the contract in case of special events like defaults or restructuring). This means that your return will be K/0 which is infinite. Clearly, the percentage difference between K/P1 with a low P1 and K/P2 with P2=0 is still big even when P1 is already low (it is the difference between a high real number and infinite).. this may help you in 2 words with no mathematics justify the fact that the Vega remains positive.
the opposite would be true for example in the unlikely case of a barrier put option that has a clause according to which the option pays K-P at maturity, but if P=0 (Knock-out barrier is touched) then the option pays nothing (I.e. an option that protects the issuer from an extreme scenario on the underlying), then yes, for some low P, the Vega would become negative.