Proof that convergent sequence in $\Bbb R$ is bounded. Convergent sequence in $\Bbb R$ is bounded. Proof: 

In the definition of a convergent sequence: 
$$(\forall \varepsilon>0), (\exists n_\varepsilon\in\Bbb N), (\forall n\in\Bbb N), ((n>n_\varepsilon)\Rightarrow(|a_n-a|<\varepsilon))$$ 
  let $\varepsilon=1$, then there exists $n_\varepsilon\in\Bbb N$ such
  that $(n>n_\varepsilon)\Rightarrow(|a_n-a|<1$). Now for
  $n>n_\varepsilon$ we have $|a_n|\leq|a_n-a|+|a|\leq 1+|a|$. Let
  $M=\max\{|a_1|,...,|a_{n_\varepsilon}|,1+|a|\}$ Then $\forall n\in\Bbb
> N, \ |a_n|\leq M$, i.e. the sequence is bounded. 

What's the idea behind this proof? I understand the first part, but then when they define $M$ I'm lost. I don't see how is that related to the first part of the proof. 
 A: The idea behind this proof is to separate the values of the sequence into 2 parts : the ones that are close to the limit (within distance 1 of it) and the ones that are "far" from the limit.
By definition of convergent sequences, those last ones are in finite number, so the set of their absolute values is bounded, by a real $M$. And the set of the others is bounded by $|l|+1$.
So the whole set of absolute values is bounded by $\max(|l|+1,M)$.
A: The idea is that after a while (i.e., after $n_\epsilon$), the sequence is bounded by $1+|a|$, and before that, the sequence is finite so it is bounded by the largest element.
What you do is you split the sequence into two parts, a starting part and "the rest"


*

*The starting part, $a_1,a_2,\dots, a_{n_\epsilon}$, is clearly bounded by $M_1=\max\{a_1,a_2,\dots, a_{n_\epsilon}\}$. It's bounded by $M_1$ because $M_1$ is the largest element of a finite set, and so it must be larger than all elements of that set.

*The rest of the sequence, $a_{n_\epsilon+1}, a_{n_\epsilon+2},\dots$, is infinite, so we can't just take its max. However, we know and can show that this part must be bounded by $M_2=1+|a|$.


Now, the first part of the sequence is bounded by $M_1$, and the second by $M_2$, so both together must be bounded by $\max\{M_1,M_2\}$, since $M\geq M_1, M_2$.

For example, you need to define $M$ because $1+|a|$ may not be the upper limit of the entire sequence. For example, the sequence $\frac2n$ has a limit of $0$, but the sequence is not bounded by $1+|0|$ since $2$ is also a part of the sequence.
However, the sequence is bounded by $\max\{2, 0+1\}=2$.
A: We want some $M>0$ so that $|a_n|\leq M$ for all $n\in\Bbb N$ and if this happens, we say that the sequence $(a_n)$ is bounded. Now, you already mentioned that
$$|a_n|<1+|a|\quad\text{for  }n=n_{\epsilon}+1,n_{\epsilon}+2,n_{\epsilon}+3,\dots.\tag{$*$}$$
What we gonna do to the numbers $|a_1|,|a_2|,|a_3|\dots,|a_{n_{\epsilon}}|$? This give an idea of  taking
$$M=\max\{|a_1|,...,|a_{n_\varepsilon}|,1+|a|\}.$$ So, using the definition of our $M$ together with $(*)$, we get
$$\begin{align}
&|a_1|\leq M\\
&|a_2|\leq M\\
&|a_3|\leq M\\
&\quad\vdots\\
&|a_{n_{\epsilon}}|\leq M\\
&|a_{n_{\epsilon}+1}|< 1+|a|\leq M\\
&|a_{n_{\epsilon}+2}|< 1+|a|\leq M\\
&|a_{n_{\epsilon}+3}|< 1+|a|\leq M\\
&\quad\vdots\\
\end{align}$$
With the above, we conclude that $|a_n|\leq M$ for all $n\in\Bbb N$.
A: See the main idea is that ,, by the convergence  property , you get a bound of an infinite part of the sequence  i.e of all the $x_n$'s after some stage.
Now then the bound becomes easy to define , just take the maximum of all the $x_n$'s before that particular $N_0$  with the bound you have got for the infinite part. Hence the bound of the sequence gets defined.
