Prove: If a sequence converges, then every subsequence converges to the same limit.
A sequence converges to a limit $L$ provided that, eventually, the entire tail of the sequence is very close to $L$. If you restrict your view to a subset of that tail, it will also be very close to $L$.
An example might help. Suppose your subsequence is to take every other index: $n_1 = 2$, $n_2 = 4$, etc. In general, $n_k = 2k$. Notice $n_k \geq k$, since each step forward in the sequence makes $n_k$ increase by $2$, but $k$ increases only by $1$. The same will be true for other kinds of subsequences (i.e. $n_k$ increases by at least $1$, while $k$ increases by exactly $1$).
For part 1, if there were a subsequence that didn't converge to the same limit, then we could find a neighborhood around the original point such that infinitely-many subsequence terms were outside the neighborhood. Then infinitely many sequence terms are outside the neighborhood, and so the sequence can't converge to the original point.
As for part 2, let me put an alternative proof forward. Suppose by way of contradiction that there is some $k$ such that $n_k<k$. We know that $n_1\geq 1$ since $1$ is the least positive integer, so we necessarily have $k>1.$ Now, $n_j<n_{j+1}$ for all $j$, so we know that the $n_j$ are all distinct, and in particular, for all $j\leq k$ we have $n_j\leq k-1$ (since $n_j\leq n_k<k$). But then $\{n_1,...,n_k\}$ is $k$-element subset of the set $\{1,...,k-1\}$, which is impossible, since a subset can't be strictly larger than a set. There's the desired contradiction.
About the indices.
$n_1\geq 1$, because $(n_k)$ is defined to be a strictly increasing sequence of indices.
Assume $n_k\geq k$ for some $k\in\mathbb{N}$, what follows by definition is that $n_{k+1} > n_k\geq k$.
If $m,n\in\mathbb{N}$ such that $m>n$, then $m\geq n+1$. (property of natural numbers)
We now conclude that $n_{k+1}\geq n_k +1\geq k+1$