Vtyjkyuk

Let $E$= every decimal digit is 4 and 7.show that it is closed and perfect?

My attempt:

$tin E'$ so $forall epsilon >0 ,exists yin E $ , $ d(y,t)<epsilon$

Claim :$tin E$


On contary assume that $tnotin E $ that is it has digits other that 4 and 7.

I wanted to show this is contradication .

But how to argue using metric that i don't know.

To show it is perfect.We have to show that no point is isolated, as already we have closedness .

On contrary say $min E$ is isolated point so there exist $r>0 $ such that $(m-r,m+r)$ contain no point of E. That there is no point other than m in it which contain 4 and 7 .It is intutionaly I know not possible because Rational are dense.But i don't know how to write above in a rigorous way.Any Help will be appreciated.

Any help will be appreciated

$E$ is closed

It is equivalent to prove that $E$ complement in $[0,1]$ is open. So take $x in [0,1] setminus E$.

Let's name $x equiv 0,x_1 x_2 dots x_n dots$ the digits of $x$.

By hypothesis, one of the digits of $x$ is not in $4, 7$. Let's say the $n$-th digit and name $overlinex_n equiv 0,x_1 x_2 dots x_n 0 dots$. Then all the reals of the open interval $(overlinex_n - 10^-n-1, overlinex_n+ 10^-n-1)$ have at least a digit not in $4, 7$: the $n$-th one is equal to $x_n$ for $y in [overlinex_n, overlinex_n+ 10^-n-1)$ and to $x_n-1 notin 4,7$ for $y in (overlinex_n-10^-n-1, overlinex_n)$.

This proves that $E$ is closed.

$E$ has no isolated point

Take $x equiv 0,x_1 x_2 dots x_n dots in E$, which means that all the $x_i$'s are in $4,7$. Then the sequence of reals $(X_m)$ where all the digits of $X_m$ are equal the one of $x$ except the $m$-th one, which is equal to $4$ if $x_m=7$ or to $7$ if $x_m=4$ is a sequence of distinct elements of $E$ which converges to $x$. Hence $E$ has no isolated point.

Your approach is valid but has led to a wall. This can be overcome but I'll offer an alternate approach. Let $xin E$ and show that it is a limit point. This can be done by writing $x=0.x_1x_2x_3...$ where $x_i_i=1^inftysubset4,7$ is a sequence. Now construct a sequence $y_i_i=1^inftysubset E$ that gets arbitrarily close to $x$ by modifying the decimal positions of $y_i$.