Partitioning an infinite set Can you partition an infinite set, into an infinite number of infinite sets?

Note: I wrote this up because it is interesting, and 'rounds out' the theory, that we can partition $X$ into an infinite number of blocks with each on being countably infinite.
I am marking this as community wiki. If several people downvote this and/or post comments that they disagree with my sentiments, I'll delete this.

Proposition 1: Every infinite set $X$ can be partitioned into blocks with each one being countably infinite.
Proof
Choose a well order $\le$ on $X$ that has no maximal elements (see this) and let $\sigma$ denote the successor function on $X$. Let $L$ denote the elements of $X$ that do not have an immediate predecessor. For each $\alpha \in L$ define
$\tag 1 L_\alpha = \{ S^n(\alpha) \, | \, \text{integer } n \ge 0\}$
This family partitions $X$ (see this) and each set must also be countably infinite. $\quad \blacksquare$.
If $L$ is finite take any $\alpha \in L$ and partition the countably infinite set $L_\alpha$ into an infinite number of blocks (see the many fine answers in this thread). So we have
Proposition 2: Every infinite set $X$ can be partitioned into an infinite number of blocks
with each one being countably infinite.
