The phrase "progressing further into" is grammatically correct and can be used in written English.
It means to continue moving or advancing deeper into something. Example: As we kept hiking, we found ourselves progressing further into the dense forest, surrounded by towering trees and chirping birds.
How do these two keep their finger on what's cool even as they progress further into adulthood?
While the U.S. had progressed further into the World Cup than England in 2014, no American man has won Wimbledon since Pete Sampras in 2000.
Up until 46 h the original water frontier progressed further into the matrix.
These tests showed that, as the specimen become worn and surface roughness increases, the contact progresses further into the mixed and boundary regime.
FinFET is a multiple-gate silicon transistor structure that nowadays is attracting an extensive attention to progress further into the nanometer era by going beyond the downscaling limit of the conventional planar CMOS technology.
Readers, as they progress further into the poem, are no doubt expected to accumulate the appropriate materials for understanding the proem as in tune with the true Epicurean message, but there is little agreement as to how this is meant to be achieved.
He triumphed as he progressed further into the role of the villain, even making it possible for viewers to support his villainous ways.
As I progress further into my transition, I feel more and more affected by the gender inequalities swirling around me, and more and more uncomfortable with the privilege I am in the process of inheriting.
As the country progresses further into the primaries we will continue to hear that defectors and outsiders cannot win a general election but the power of trust should not be discounted and may be the determining factor of our next president.
