Nash equilibrium of sequence of games
This is just an extended remark:

Let $\Delta$ be the space of Borel probability measures on $[0,1]$. An equilibrium can be represented as an element of $\Delta^2=\Delta\times\Delta$. A two player game with continuous payoff-functions with action spaces $[0,1]$ can be viewed as an element of $C[0,1]^2\times C[0,1]^2$, where $C[0,1]^2$ is the space of continuous functions (here representing payoff functions) from $[0,1]^2=[0,1]\times[0,1]$ to the reals, endowed with the topology of uniform convergence, given by the $\sup$-norm $\|\cdot\|_\infty$ defined by
$$\|f\|_\infty=\sup_{x\in [0,1]} |f(x)|=\max_{x\in [0,1]} |f(x)|.$$
One defines $C[0,1]$ similarly.

There is also a well-behaved metrizable compact topology on $\Delta$, the topology of weak convergence (or weak*-topology); the coarsest topology such that the function $\mu\mapsto\int f~\mathrm d\mu$ is continuous for each $f\in C[0,1]$. In this context, convergence in this topology is convergence of cumulative distribution functions at continuity points. We let $\Delta^*$ be the set of Borel probability measures on $[0,1]\times [0,1]$, similarly endowed with the topology of weak convergence.

We can then define an equilibrium correspondence $$\eta:C[0,1]^2\times C[0,1]^2\to 2^{\Delta\times\Delta}$$
such that $\eta(v,u)$ is the set of equilibria given the payoff-functions $u$ and $v$. The correspondence has nonempty values by a theorem of Glicksberg. Since $\Delta\times\Delta$ is compact, we can show that $\eta$ is upper hemicontinuous by showing it has a closed graph.  The function $p:\Delta\times\Delta\to\Delta^*$ that maps mixed strategies to their independent product is continuous (see Billingsley's book on weak convergence). It follows, using a bit of analysis, that the "expected payoff"-function $k_i:\Delta\times\Delta\times C[0,1]^2\to\mathbb{R}$ is continuous for $i=1,2$. Now for each $\mu\in\Delta$, let 
$$E_\mu^1 \{(v,u,\nu,\tau)\in \Delta\times\Delta\times C[0,1]^2\times C[0,1]^2\mid k_1(\nu,\tau,v)\geq k_1(\mu,\tau,v)\}$$
and
$$E_\mu^2 \{(v,u,\nu,\tau)\in \Delta\times\Delta\times C[0,1]^2\times C[0,1]^2\mid k_2(\nu,\tau,u)\geq k_1(\nu,\mu,u)\}.$$ Each of these sets is closed and the graph of $\eta$ is simply
$$\bigcap_{\mu\in\Delta}E_\mu^1\cap \bigcap_{\mu\in\Delta}E_\mu^2,$$
a closed set, so $\eta$ is upper hemicontinuous.

Now this doesn't tell us anything about what happens if payoff functions converge pointwise and not uniformly!!! The only useful information is that strategy spaces are compact, so there will still be a convergent subsequence. If you can show that this subsequence must converge to your candidate equilibrium- this will require making use of the details of the game, you should be good to go.