Elon is a start-up entrepreneur in the Silicon Valley. Every year he starts a new company and sells it at the end of the

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Elon is a start-up entrepreneur in the Silicon Valley. Every
year he starts a new company and sells it at the end of the year.
If things go well, he gets $1 billion, and if things go wrong, he
loses $1 billion. The probability of both possible outcomes is
equal. Since he keeps starting companies in other sectors, we
assume that the outcomes of the different start-ups are
independent. At the beginning of each year, he must pay (1−ρ)·100%
of his wealth to the tax authorities, where 0 ≤ ρ < 1. If his
wealth is negative, he will receive (1−ρ)·100% of his debt as state
aid. After all, because Elon has frozen himself to absolute zero
and communicates with the outside world via a neural link, we can
assume that he will live forever. Let Xn be Elon's net worth at the
end of year n, expressed in billions of dollars. Assume that Elon
starts with no wealth at the beginning of year 1.
Show that lim n→∞ E(Xn) = 0 and lim n→∞ Var(Xn) = 1 /
(4 − 4ρ^2) .