Loop example in distance vector protocol that uses the split horizon
technique.

Graph: Destination 1.  Nodes 2, 3, and 4 are connected in a triangle.
Edge lengths (2,3) = 1, (3,4) = 1, and (4,1) = 2.  We also have edges
(2,1) = L and (4,1) = 1.

So the SPT is 2->3->4->1.

Suppose edge (4,1) breaks.  Then 4 sets its distance to 1 to be inf.
It broadcasts this information to both 3 and 2.  Meanwhile 2 sends
message to 4 saying that its distance to 1 is 3.  4 updates its
distance to be 5 and forwards this information to 3.  3 updates its
distance to be 6 and forwards this to 2.  2 updates its distance to be
7 and forwards this to 4.  This cycle continues.

How does DSDV avoid loops?

DSDV uses sequence numbers associated with each destination.  Nodes
exchange distance vectors as usual, but now a node u updates its
next-hop and/or distance in the following cases:

-- if the new info has a newer sequence number, then the distance is
changed accordingly.  if the distance is inf, then next-hop is set to
nil.

-- if the new info has an old sequence number, then this is ignored.

-- if the new info has the same sequence number but a smaller
distance, then the distance is changed accordingly.

Proof of loop-freedom.

The claim is that all points of time, the edges (i, p(i)) -- where
p(i) is the parent of i -- form a forest, with the root of each tree
being either the destination or a node that has p(i) = NIL.
Furthermore, the sequence number at a root with p(i) = NIL is higher
than any of its children.

The proof is by induction on the number of events.  When a link (u,v)
breaks and v is u's next-hop, then u sets its next-hop to NIL and
increases the sequence number.  This action is going to maintain the
forest property and also ensure that the sequence number of root is
higher than its children.

When a node receives new information from a parent node, it updates
its information accordingly, adopting the new sequence number and
setting its parent to nil if the parent's distance is infinity.

When a node receives information from another node, it checks the
sequence number.  If the distance is better than currently or sequence
number is new, it updates the distance and new next-hop, maintaining
the invariant.