Turtle crawling on elastic cable

[SataSata]

Homework Statement

A turtle is crawling along a cable made of super elastic material. The turtle speed is constant at 1mm per second. The cable's length is 1km and at the same time it is being stretched at a constant speed of 1km per second. Suppose the turtle is immortal and the cable can be stretched infinitely long, will the turtle reach the other end of the cable? If so, how long will it take?
The stretching of the cable also causes the turtle to move together with the point it's at.

Homework Equations

The Attempt at a Solution

Cable length over time l(t) = Initial length + cable length growth = 1 + 1t

Turtle position over time x(t) = Initial speed over time + cable length growth = 0.001t + 1t = 1.001t

For turtle to reach the end:
x(t) > l(t)
1.001t > 1 + t
t > 1000s
Hence turtle will reach end of cable in 1000s.

So what I did is actually wrong. Can somebody explain why?

At 10s, cable length is 11km while turtle position is at 10.01km
At 100s, cable length is 101km while turtle position is at 100.1km
At 1000s, cable length is 1001km while turtle position is also at 1001km
Why is this wrong? Did I misunderstood how the turtle's position will change with the stretching of the cable?

 

[BvU]

Turtle doesn't advance with the cable length increase of the whole cable. Only with the increase of the piece behind him/her (how do you distinguish a female turtle from a male one ?)