Are strict dominance solvable games weakly dominance solvable?
You're right. Let $s^*=(s_1^*,\dots,s_N^*)$ be the equilibrium of a strictly dominance solvable game. Then by definition, 
$$
u_i(s_i^*,s_{-i})&gt;u_i(s_i,s_{-i})
$$
for all $i$, all $s_i\ne s_i^*$ and all $s_{-i}$. This implies that 
$$
u_i(s_i^*,s_{-i})\ge u_i(s_i,s_{-i})
$$
for all $i$, all $s_i\ne s_i^*$, all $s_{-i}$ and with strict inequality for at least some $s_i$ (in fact, for all $s_i\ne s_i^*$). This makes $s^*$ an equilibrium satisfying the weak dominance solvability criterion.



Your quoted definition of strict/weak dominance solvability makes me a little uncomfortable though. I'd say 


  A game is strict (or weak) dominance solvable if the process of iteratively removing strictly (or weakly) dominated strategies leads to a unique outcome (i.e. only one strategy for each player survives).