What is the proof for the conditions of a normal approximation of the binomial distribution: $np > 5$ and $n(1 - p) > 5$? For a normal approximation of the binomial to be valid,
$np > 5$
$n(1 - p) > 5$
What is the algebraic proof for this and where does the $5$ come from?
 A: As @JeanMarie Comments, there is no proof. This is a very rough rule-of-thumb. I suppose it survives
because it is considered 'easier to remember' than better criteria. (And it is a little
better if you use 10 instead of 5.)
You can see a discussion of several better 'rules' in the Wilipedia
article on 'binomial distribution' under normal approximation.
Especially
for small $n,$ a major consideration is that normal approximation
to normal works best for $p$ near 1/2, so that the binomial is
nearly symmetrical, and thus easier to approximate with the 
symmetrical normal.
One of the easiest to remember of the better rules is to have both
$np/q > 9$ and $nq/p > 9,$ where
$q = 1-p.$ This ensures that binomial values in $[0,n]$ correspond
to standard normal values in $(-3,3)$.
With the current wide availability of statistical software, there is
seldom a reason in practice to use a normal approximation.
For example, if $X \sim Binom(100, 0.1),$ then the exact
value of $P(X \le 5) = 0.0576$ can be obtained in R statistical
software as follows:
pbinom(5, 100, .1)
## 0.05757689

But about the most accurate normal approximation with the usual
method (including continuity correction) is 0.0668, obtained by:
n = 100;  p = .1;  q = 1-p
mu = n*p;  sg = sqrt(n*p*q)
pnorm(5.5, mu, sg)
## 0.0668072

Generally, it is not prudent to expect more than two-place
accuracy for a normal approximation, although there are even cases that
violate rules of thumb and still give surprisingly good results. Consider
$P(X = 2) = P(1.5 < X \le 2.5) = 3/8 = 0.3750,$ with $X \sim Binom(3, 1/2)$.
The normal approximation with continuity correction gives 0.3759. 
n = 3;  p = .5;  q = 1-p
mu = n*p;  sg = sqrt(n*p*q)
diff(pnorm(c(1.5,2.5), mu, sg))
## 0.3758935

Reference: J. Pitman: Probability, Springer, 1993 gives a more
sophisticated and accurate method of normal approximation than the
one in general use.
