Is it true that a sequence $b_n$ has an accumulation point $+\infty$ iff it is not bounded from above 
Let $(b_n)_n$ be a real valued sequence. I claim that $(b_n)_n$ has the accumulation point $+\infty$ iff $(b_n)_n$ is not bounded from above.

My prove would be the following.
$\Rightarrow$ Let me assume $(b_n)_n$ has the accumulation point $+\infty$. This means that for infinitely many $n\in \Bbb{N}$, $b_n\geq C$ for all $C\in \Bbb{R}$. But then $b_n$ is not bounded from above.
$\Leftarrow$ Let me assume $(b_n)_n$ is not bounded from above, i.e. $\forall C \in \Bbb{R}$ there exists $n\in \Bbb{N}$ such that $b_{n}\geq C$. Now let me pick a sequence $C_k$ in $\Bbb{R}$ such that $C_k\rightarrow \infty$ as $k\rightarrow \infty$. Then for each $k$ I can find $n_{k}$ such that $b_{n_{k}}\geq C_k$ now consider the subsequence $(b_{n_k})_k$ of $(b_n)_n$ then we see that this converges to $+\infty$ so $+\infty$ is a accumulation point of $(b_{n_k})_k$ and hence also of $(b_n)_n$.
Does this work like this or did I make a mistake?
 A: First note you can make life easier by choosing $C_k=k.$
Next, to correctly form a subsequence, you need to ensure that $n_k > n_{k-1}$ always holds,  i.e. that the selected $n$ values don't go in reverse. (See nice remarks here: https://math.stackexchange.com/a/3204170)
For example, for $C_k=k=1$, you know there is some $n_1: b_{n_1} \geqslant 1$, and this might be $n_1 = 10$. For $k=2$, you know there is some $n_2: b_{n_2} \geqslant 2$, but this might be $n_2 = 8$.
This is not a big problem (some may say a technicality), but you should note that if a sequence $b_n$ is unbounded above, then so is any tail of the sequence. (Prove this.)
The upshot is that you can choose at each stage an $n_k > n_{k-1}$ such that $b_{n_k} \geqslant k$.
I think your last step is alright, but you could also now simply use your definition: let $C$ be real, arbitrary. We need to show that for infinitely many indices, $b_n \geqslant m$ holds with $ \max(1,C) < m,$ an integer. These indices are: $$n_m,\;n_{m+1},\;n_{m+2},\;\dots.$$
