Show that an equilibrium in strictly dominant strategies is a unique Nash equilibrium
It will be useful to spell out the relevant definitions.

Let $A_i$ be the set of possible actions for player $i$, $A_{-i}$ be the set of possible joint actions of all players except for player $i$, $A$ be the set of possible joint actions of all the players. For $a=(a_i,a_{-i})\in A$, let $u_i(a)$ denote the payoff to player $i$ from action $a_i$, given that the other players play $a_{-i}$.

Definition 1: An action $a_i\in A_i$ is weakly dominant for player $i$ if for every $a_{-i}\in A_{-i}$, $u_i(a_i,a_{-i})\ge u_i(\overline{a_i},a_{-i})$ for every $\overline{a_i}\in A_i$, i.e. no matter what the other players do, action $a_i$ yields a payoff at least as high as any other action available to player $i$. [Strict dominance given by $&gt;$ instead of $\ge$.]

Definition 2: An action $a_i\in A_i$ is a best response to action $a_{-i}\in A_{-i}$ if $u_i(a_i,a_{-i})\ge u_i(\overline{a_i},a_{-i})$ for every $\overline{a_i}\in A_i$, i.e. If I fix other players' actions to some particular joint action, then $a_i$ yields a payoff to player $i$ at least as high as any other action available to player $i$.

Claim 1: If an action $a_i\in A_i$ is strictly or weakly dominant, then it is a best response for any joint action $a_{-i}\in A_{-i}$. [This follows from Definitions 1 and 2 immediately.]

Definition 3: A joint action $a\in A$ is a Nash equilibrium if for each player $i$, action $a_i$ is a best response to $a_{-i}$.

Claim 2 (your question): A strategy profile (i.e. joint action) $s=(s_1,...,s_n)\in A$ in which each $s_i$ is a (strictly or weakly) dominant strategy is a Nash equilibrium.

Note that Claim 2 is a consequence of Claim 1.