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What I need to show: Equip each $X_n$ with the discrete topology, give $\prod_{n\geq 1}^{}X_n$ the product topology and give $\lim\limits_{\leftarrow n} X_n$ the corresponding subspace topology. Explain why this topology does not have to be the discrete topology.

So I'm not sure if I have understood this correctly. Do I need to show that the subspace topology generated by the set $\lim\limits_{\leftarrow n} X_n$ doesn't necessarily have to be the discrete topology on $\lim\limits_{\leftarrow n} X_n$? I'm not quite sure how to show it..

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Yes, you’re supposed to show that $\varprojlim X_n$ need not be a discrete subspace of $\prod_nX_n$. HINT: For each $n\in\Bbb Z^+$ let $X_n=\{0,1\}^n$ with the discrete topology. For $m\le n$ let $p_{m,n}:X^n\to X^m$ be the projection to the product of the first $m$ factors of $X^n$. Show that $\varprojlim X_n$ is homeomorphic to $\{0,1\}^{\Bbb Z^+}$, the Cantor space, whose topology is certainly not discrete.

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    I might be wrong, but I think that giving an example is not what I'm supposed to in this. I think I have to show it in terms of the product topology satisfies that the opens in $\prod_{n\geq 1}^{}X_n$ is $\prod_{n\geq 1}^{}U_n$ where $U \subsetneq X$ for only finitely many i's. – 2017-01-12
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    @seht111: You have to give an example, and the one that I suggested is one of the simplest. Some inverse limits of a sequence of discrete spaces do have the discrete topology, so in order to show that such an inverse limit need not have the discrete topology, you must produce an example in which its topology is not discrete. Of course the product topology on $\prod_nX_n$ is involved: by definition the inverse limit is a subspace of that product. – 2017-01-12
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The inverse limit is a special subspace of the infinite product of the spaces involved. An infinite product of discrete spaces is not discrete. In fact, all zero-dimensional separable metric spaces can be seen as subspaces of $\{0,1\}^\mathbb{N}$, and many of these are not dicrete (like the irrationals, and the rationals). All of these can be the limits of inverse systems of discrete spaces.