Limit points of the set ${frac {varphi(n) }n : nin mathbb{N}}$

Let $varphi(n)$ denotes the cardinality of the set ${a | 1le ale n, (a,n)=1}$ where $(a , n)$ denotes the gcd of $a$ and $n$. Which of the following is NOT true ?

$(a)$ There exist infinitely many $n$ such that $varphi (n)gt varphi(n+1)$

$(b)$ There exist infinitely many $n$ such that $varphi (n)lt varphi(n+1)$

$(c)$ There exist $Nin mathbb{N}$ such that $Ngt 2$ and for all $ngt N$, $varphi (N)lt varphi(n)$

$(d)$ The set ${ frac {varphi (n)}{n} : nin mathbb{N} } $ has finitely many limit points.

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My Thinking:

$(a)$ is true by selecting $n=p$ ( an odd prime greater than $3$)

Justification:

Since $p$ is odd, $p+1$ is even. Now $varphi(p)=p-1$ and $varphi(p+1)le p+1-frac{p+1}{2}=frac{p+1}2$

So $varphi(p+1)le frac{p+1}2 lt p-1=varphi(p)$

By the infinitudes of primes, the result follows.

$(b)$ Is true by taking $n=p-1$ where $p$ is an odd prime greater than $3$ and following the same line of reasoning.

$(c)$ This statement (if true ) means that the set ${varphi(n) : ngt N}$ is bounded below.

I think it is true

My argument :

Let $N=6$ . Then $varphi(6)=2$

If $ngt 6$ and is prime , then clearly $varphi(n)gt 2$

Otherwise, if $n$ is composite, then let

$n=p_1^{r_1} p_2^{r_2}…p_k^{r_k}$ .

$varphi(n)=p_1^{r_1-1}p_2^{r_2-1}…p_k^{r_k-1}(p_1-1)(p_2-1)..(p_k-1)$

If $phi(n)=2$ , then the distinct primes can be just $2$ and $3$. So the above equality reduces to

$2=phi(n)=2^{r_1}3^{r_2-1}$ when both the factors $2$ and $3$ are present .

This gives $r_1=r_2=1$ and so $n=6$

When only $3$ present , then we get $n=3$ and if the factor is only $2$ , then $n=4$.

So the result follows as $N=6$ works

$(d)$ This is NOT true.

Justification :- Let ${p_i}$ be the enumeration of primes.

Let an index $k$ be fixed and $i$ be the running index.

Then if the only prime factors of $n_i$ are $p_k, p_i$ , then
$frac{varphi(n_i)}{n_i}=big( 1-1/p_kbig)big(1-1/p_ibig)$

So ,letting $ito infty$, we have

$lim_{ito infty}frac{varphi(n_i)}{n_i}=1-1/{p_k}$

Since $p_k$ is arbitrary, there are infinitely many limit points (maybe uncountably many) limits points of the given set.

Please check if my work is correct or the answers can be shortened in any way.

Thanks for your time.