p.s Are our observations of the world inductive, or is it just the conclusions that we draw from them which are inductive?
Proving whether 1) we ourselves exist and proving whether 2) an independent world exists are two distinct but related questions.
The most influential answer to the first question is Descartes': cogito ergo sum, or "I think before I am". Various critics, including Nietzsche and Kierkegaard among others, have pointed out that the meaning of "I" here is critical and not clearly addressed. But the experience of doubt necessarily implies the existence of some subjective entity, the doubter.
As for the second question, the existence of a world independent from the thinker essentially amounts to materialism, as opposed to idealism. The fundamental problem with strong forms of idealism is that the many things (our apparent ability to communicate with other people, the regularity of observed physical phenomenon, etc.) are difficult to explain, short of some far-fetched "brain in a vat" scenario.
To feel whether we or the world exists, I don't think a child born in this world needs to know about inductive logic or deductive logic. It doesn't wish to prove or even to say whether anything exists. It is because of our 'ignorance' we think so. Not they, actually we are ignorant. [They know that it is nonsense to say so using a mortal body :)] So there is no need to prove it. It is like an axiom in almost all persons' case.
To say that something exists there must be a second thing. If the second thing is another form of the first and both of them have no permanent existence in this world, we can't even use the word 'exist'. But we should use it for our daily activities. The Ultimate Truth is beyond existence and nonexistence.
So I can't find the significance of your question from another substratum.
It is becoming alarmingly plausible that the universe is a simulation, or even the projection of a simulation. The universe appears to pop into existence when we look at it, and not before, which would be the logical way to simulate something as vast and complex as the universe. Everything from the macroscopic action of gravity over vast distances, to the nature of quantum reality at the very small scale seems to suggest simulation. There is also some evidence (relating to the expansion of spacetime) that the events of deep time were simulated at a faster pace than current events. In other words we are more important to the simulation than primordial life. Or we take more power to simulate and thus the simulation increasingly lags. If the universe is a simulation it may be possible to communicate with the construct itself (speak to God).