Do we survive bodily death?
There are reasons to think so, but nothing that qualifies as proof. Perhaps the most sensible attitude (because it’s mine) is that if we do survive death, then we’ll find out someday. If we don’t survive death, then we’ll never know, so it won’t bother us.
A kind of proof was offered by Moses Mendelssohn (1729-1786), the German Jewish philosopher whose fame was such that he was nicknamed “the Socrates of Berlin.”
However, Mendelssohn seemed so distressed by the subject of death that it reminded me of another nickname: “Gloomy Gus,” for people who always look at the negative side of life.
He wrote about the issue in his 1767 book Phaedo, or Concerning the Immortality of the Soul. “Phaedo” refers to a dialogue by the ancient Greek philosopher Plato, in which Socrates, as he is dying, discusses imortality with his students.
Mendelssohn has two main arguments for the immortality of the soul. The first is metaphysical, but since it’s both complicated and unconvincing, I won’t inflict it on you. Immanuel Kant gives a fairly good summary of the argument in his Critique of Pure Reason (B-413).
The second argument is where Mendelssohn turns into a Gloomy Gus of major proportions. The argument isn’t very convincing either, but at least it’s interesting and you can understand what he’s talking about.
He seems to worry about death a lot more than a psychologically healthy person should:
"During happy times, the dreadful thought of nonexistence winds its way through the most delightful representations like a snake through flowers, and poisons the enjoyment of life. During unhappy times, it dashes a man to the ground in complete hopelessness ..." 
His basic argument, as far as I can tell, is this:
1. If death is the end of us, then life is our ultimate good.
2. If life is our ultimate good, then we have a right to do whatever is necessary to protect it.
3. Each country has a right to demand its citizens sacrifice their lives for their country.
4. Our right to life conflicts with our countries’ right to demand that we sacrifice our lives.
5. There are no irresolvable conflicts of rights.
6. Therefore, death is not the end of us.
In this argument, Mendelssohn deduces a metaphysical conclusion (the soul is immortal) from a moral conundrum: if the soul isn’t immortal, then there are conflicts of rights.
The problem is that all of his premises are highly debatable. He might believe in them, but they require a lot more argument to convince anyone else.
I’m not sure that Mendelssohn himself thought his argument was valid. He might have thought that belief in immortality promoted happiness and morality, so it was justified for him to use heavy-breathing rhetoric to convince people it was true.
In his “Open Letter to Lavater” (1769), he said that when truth conflicts with the social good, we should sometimes support the social good instead of the truth:
"Whoever cares more for the welfare of mankind than for his own renown will keep a rein on his opinions concerning ... erroneous religious opinions that are accidentally connected to the promotion of the good." 
Mendelssohn seems to have been a wise man: indeed, wiser than we might think from reading his shaky argument in favor of human immortality.
Gottlieb, M., editor (2011), Moses Mendelssohn: Writings on Judaism, Christianity, and the Bible. Lebanon: Brandeis University Press.
1. Gottlieb, M. (2011), p. 246.
2. Ibid, p. 12.