real analysis - Prove the curve $(x(t),y(t))$ is contained and bounded within a larger square centered at the origin

I'm analyzing continuous functions on compact intervals and came up with this question from Arthur Mattuck - Introduction to analysis book. It exactly says:

Let $x(t)$ and $y(t)$ be continuous on $[a,b]$. As $t$ varies over this interval the point $(x(t),y(t))$ traces out a curve in the $x-y$ $plane$.Prove this curve is contained within some large square centered at the origin and show by example that this might be not true if the interval used in the $x-axis$ is of the form $(a,b)$

My approach:

1. Compactness implies a closed and finite interval. Excluding one of them, then this is not compact and therefore not bounded.




2. Also that a curve is given by $\sqrt{x^2+y^2}$, but not sure if this is useful.

I don't know how to go over this. If I have to proof by contrapositive or what else. Any help would be appreciated.