Yes. This is called "combined arms" and occurred all the time; providing you could afford the cost of horses and the delay of infantry.

While the range and reach of pole-arms had some uses before the arrival of set-piece battles involving horses (a/k/a The Cavalry); pole-arms exploded in popularity once horses were common enough and cheap enough - especially the Halberd as you could hook the uppity noble off his horse and then stab repeatedly with a dagger.

So unless you had a mass of archers (horseback or otherwise); a pre-firearm land battle would need infantry as soon as you were bottled up in terrain that limited manoeuvrability. Though one could argue that a desire to destroy the enemy head-on and an aristocrat distaste of guerrilla warfare induced more set-piece battles than would otherwise occur.

Conscripted soldiers often had no say on whether to pursue asymmetric warfare, hence battles such as the "Battle of Hamburger Hill" in Vietnam or the Battle of Towton in the War of the Roses.

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Yes. This is called "combined arms" and occurred all the time; providing you could afford the cost of horses and the delay of infantry.

While the range and reach of pole-arms had some uses before the arrival of set-piece battles involving horses (a/k/a The Cavalry); pole-arms exploded in popularity once horses were common enough and cheap enough - especially the Halberd as you could hook the uppity noble off his horse and then stab repeatedly with a dagger.

So unless you had a mass of archers (horseback or otherwise); a pre-firearm land battle would need infantry as soon as you were bottled up in terrain that limited manoeuvrability. Though one could argue that a desire to destroy the enemy head-on and an aristocrat distaste of guerrilla warfare induced more set-piece battles than would otherwise occur.

Conscripted soldiers often had no say on whether to pursue asymmetric warfare, hence battles such as the "Battle of Hamburger Hill" in Vietnam or the Battle of Towton in the War of the Roses.

Elaborating on the nature of cavalry*: Fighting from horseback essentially added extra manoeuvrability to combat - a way of increasing your options of engagement and decreasing the enemy's as per Sun Tzu's chapter on movement. Cultures we consider did very well without massed infantry (Mongol Golden Horde, Parthian Horse Archers, Polish Winged Hussars) made sure to avoid meat-grinding close-quarter combat. Extensive use of weapons with asymmetric reach (archery, thrown spears, really long disposable lances) plus the rider's elevated field of vision (improved battle awareness) also helped evade the enemy's projection of force and enhance your own.

Infantry wasn't stupid; their aim was to either restrict the horses (selection of terrain) or kill the horse and rider at a distance (massed archery). Selection of terrain limited range of environments cavalry could fight in (swampland, mountains, cities, etc) and completely ruled out holding on to hostile territory without infantry as the local populace could re-shape/the/terrain to be unsafe for cavalry. Cultures famed for cavalry either had infantry (Poland, Parthia) or were nomads with few fixed settlements (Mongols**).

_{* A Military Stack Exchange is still in the Area 51 staging zone.
** The Mongol "Empire" dissolved quickly. In so far as its fragments had internal cohesion, this was because they were simply the original nations under new management - with local culture, laws and infantry.}