The simplest layman's explanation I can think of in terms of a counter-example.

You know what a step is, right? Something that looks like, well, a step on a staircase.

Now imagine a smooth ramp, like a handicap access ramp, that gets you between one level to another. Let's say that to qualify as a "ramp" the thing can't be vertical.

Now imagine making that ramp progressively steeper. At each iteration you make it a bit steeper, but don't make it vertical. One way to do this is to increase the steepness at each step by half the remain angle to the vertical. So if you are at 60 degrees, you go to 75, if you are at 75, you go to 82.5 and so on.

Obviously, if you keep doing it an infinite number of times, you get closer and closer to a vertical "step" in the sense of a staircase. But you never quite get there.

Now think about the "space" of "increasingly steep ramps" you've created.

This space of increasingly steep ramps is NOT a Hilbert space, because the end state, the vertical step, is not a ramp at all.

This is what is meant by the technical idea of a "complete infinite sequence" ... it converges to a point IN the space. In this case, what we call "continuous functions."