Assume $T_n,T$ are bounded bijective linear operators $T_n to T$ pointwise. Show $T_n^{-1}to T^{-1}$ pointwise $iff$ $|T_n^{-1}|leq C$

Assume $T_n,T$ are bounded bijective linear operators $X to Y$ and $T_n to T$ pointwise. Show $T_n^{-1}to T^{-1}$ pointwise $iff$ $|T_n^{-1}|leq C$

Note: $X,Y$ are banach spaces.

My proof:

Forward direction is uniform boundedness principle. Backwards:

Let us assume that $T_n^{-1}(y)not to T^{-1}(y)$ so there is a subsequence and $y$ s.t $|T_{n_k}^{-1}(y)- T^{-1}(y)|geq epsilon$. Now we know that $T_{n_k}(x)to T(x)$ and since $T^{-1}_{n}$ are uniformly boundaed we get that $|T^{-1}_{n_k}(T_{n_k}(x)-T(x))|<C|T_{n_k}(x)-T(x)|to 0$ Now just let $x=T^{-1}(y)$ and we arrive at contradiction. Is this correct? It seems roundabout in my opinon and there is probably more direct way to do it.