Let $X$ be the topological space and $U_i$ are open set.If $U_i\subset U_{i+1}$ and $\cup^{\infty}_{i=1}U_i=X$. How can I prove that if infinitely many $j$, $i$-th homology vanishes $H_i(U_j)=0$, then $H_i(X)=0$?

Let $X$ be a topological space and $U_i$ open subsets. If $U_i\subset U_{i+1}$ and $\bigcup^{\infty}_{i=1}U_i=X$. How can I prove that if for infinitely many $j$, the $i$-th homology vanishes $H_i(U_j)=0$, then $H_i(X)=0$?

Let $X$ be the topological space and $U_i$ are open set.If $U_i\subset U_{i+1}$ and

Let $X$ be the topological space and $U_i$ are open set.If $U_i\subset U_{i+1}$ and $\cup^{\infty}_{i=1}U_i=X$. How can I prove that if infinitely many $j$, $i$-th homology vanishes $H_i(U_j)=0$, then $H_i(X)=0$?

Let $X$ be the topological space and $U_i$ are open set.If $U_i\subset U_{i+1}$ and $\cup^{\infty}_{i=1}U_i=X$. How can I prove that if infinitely many $j$, $i$-th homology vanishes $H_i(U_j)=0$, then $H_i(X)=0$?

Let $X$ be the topological space and $U_i$ are open set.If $U_i\subset U_{i+1}$ and