I have one peculiar doubt regarding monotone sequences. I came across two sequences which are $a_n=\frac{5^n}{n!}$ and $b_n =\sqrt[n]{n}$
We know that a sequence is strictly decreasing if $a_{n+1}<a_n$, $\forall n \in \mathbb{N}$. Now for these two sequences, they actually start decreasing for $n \geq 5$ and $n \geq 3$ respectively. So my first question is:
$Q.1$ Will it make more sense if we define a sequence as strictly decreasing if $\exists k \in \mathbb{N}$, such that $a_{n+1}<a_n$, $\forall n \geq k$
Now coming to Question $2$, if now i consider an sequence whos terms are $$2,\frac{1}{2},\frac{2}{3},\frac{3}{4},....$$ which is quite obvious that the sequence is strictly increasing from the second term. By Monotone convergence theorem we know that an increasing bounded sequence converges to its supremum.But the supremum of the above sequence is $2$ and infact its converging to $1$.
$Q.2$ Again will it make more sense if we define a sequence as strictly increasing if $\exists k \in \mathbb{N}$, such that $a_{n+1}>a_n$, $\forall n \geq k$
So are we simply discarding the first few terms?