The following statement is being used a lot in the literature, and I wonder how to prove it.

Let $M$ be an infinite dimensional von-neumann algebra (with unit element), show that there is an increasing series of projections {$p_i$}$_{i=1}^\infty$ such that $\forall n : p_n \neq 1$ and $p_n\to 1$.

My thoughts so far:

I know that in an infinite dimensional von-neumann algebra there is an infinite set of pairwise orthogonal projections. so taking their partial sums would be an increasing series of projections, however i'm not sure if they converges to 1.

Thanks in advance!

The following statement is being used a lot in the literature, and I wonder how to prove it.

Let $M$ be an infinite-dimensional von Neumann algebra (with unit element), show that there is an increasing series of projections {$p_i$}$_{i=1}^\infty$ such that $\forall n : p_n \neq 1$, and $p_n\stackrel{n\to\infty}\longrightarrow 1$.

My thoughts so far:

I know that in an infinite-dimensional von Neumann algebra there is an infinite set of pairwise orthogonal projections. so taking their partial sums would be an increasing series of projections, however I'm not sure if they converge to 1.

Thanks in advance!