Fibonacci sequence in the Pascal triangle

Where is the Fibonacci sequence in the Pascal triangle?

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1

Count as follows.
Start left with a 1.
While on a number:
- Move right
- Move up
- Add to running total.

Let's go.
First row.
1.
Stop.

Second row.
1
Stop.

Third row.
1, right, up
1
Stop.

Fourth row
1, right, up
2
Stop.

Fifth row
1, right, up
3, right, up
1
Stop.

Sixth row
1, right, up
4, right, up
3
Stop.

Totals were 1, 1, 2, 3, 5, 8.

Fibonacci sequence, obvious.

Now what's the proof?